Calculus Examples

$lim_{x \to 0} {(\frac{\sin (x)}{x})}^{\frac{1}{x^{2}}}$

Step 1

Use the properties of logarithms to simplify the limit.

Tap for more steps...

Step 1.1

Rewrite ${(\frac{\sin (x)}{x})}^{\frac{1}{x^{2}}}$ as $e^{\ln ({(\frac{\sin (x)}{x})}^{\frac{1}{x^{2}}})}$ .

$lim_{x \to 0} e^{\ln ({(\frac{\sin (x)}{x})}^{\frac{1}{x^{2}}})}$

Step 1.2

Expand $\ln ({(\frac{\sin (x)}{x})}^{\frac{1}{x^{2}}})$ by moving $\frac{1}{x^{2}}$ outside the logarithm.

$lim_{x \to 0} e^{\frac{1}{x^{2}} \ln (\frac{\sin (x)}{x})}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 0} \frac{1}{x^{2}} \ln (\frac{\sin (x)}{x})}$

Step 2.2

Combine $\frac{1}{x^{2}}$ and $\ln (\frac{\sin (x)}{x})$ .

$e^{lim_{x \to 0} \frac{\ln (\frac{\sin (x)}{x})}{x^{2}}}$

Step 3

Apply L'Hospital's rule.

Tap for more steps...

Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

Tap for more steps...

Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$e^{\frac{lim_{x \to 0} \ln (\frac{\sin (x)}{x})}{lim_{x \to 0} x^{2}}}$

Step 3.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 3.1.2.1

Move the limit inside the logarithm.

$e^{\frac{\ln (lim_{x \to 0} \frac{\sin (x)}{x})}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.2

Since $\cos (x) \leq \frac{\sin (x)}{x} \leq 1$ and $lim_{x \to 0} \cos (x) = lim_{x \to 0} 1 = 1$ , apply the squeeze theorem.

$e^{\frac{\ln (1)}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.3

The natural logarithm of $1$ is $0$ .

$e^{\frac{0}{lim_{x \to 0} x^{2}}}$

Step 3.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 3.1.3.1

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$e^{\frac{0}{{(lim_{x \to 0} x)}^{2}}}$

Step 3.1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{0}{0^{2}}}$

Step 3.1.3.3

Raising $0$ to any positive power yields $0$ .

$e^{\frac{0}{0}}$

Step 3.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{0}{0}}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{0}{0}}$

Step 3.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{\ln (\frac{\sin (x)}{x})}{x^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\frac{\sin (x)}{x})]}{\frac{d}{d x} [x^{2}]}$

Step 3.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\frac{\sin (x)}{x})]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{\sin (x)}{x}$ .

Tap for more steps...

Step 3.3.2.1

To apply the Chain Rule, set $u$ as $\frac{\sin (x)}{x}$ .

$e^{lim_{x \to 0} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{\sin (x)}{x}]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$e^{lim_{x \to 0} \frac{\frac{1}{u} \frac{d}{d x} [\frac{\sin (x)}{x}]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.2.3

Replace all occurrences of $u$ with $\frac{\sin (x)}{x}$ .

$e^{lim_{x \to 0} \frac{\frac{1}{\frac{\sin (x)}{x}} \frac{d}{d x} [\frac{\sin (x)}{x}]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{x}$ .

$e^{lim_{x \to 0} \frac{1 \frac{x}{\sin (x)} \frac{d}{d x} [\frac{\sin (x)}{x}]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.4

Multiply $\frac{x}{\sin (x)}$ by $1$ .

$e^{lim_{x \to 0} \frac{\frac{x}{\sin (x)} \frac{d}{d x} [\frac{\sin (x)}{x}]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.5

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (x)$ and $g (x) = x$ .

$e^{lim_{x \to 0} \frac{\frac{x}{\sin (x)} \cdot \frac{x \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [x]}{x^{2}}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.6

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{lim_{x \to 0} \frac{\frac{x}{\sin (x)} \cdot \frac{x \cos (x) - \sin (x) \frac{d}{d x} [x]}{x^{2}}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{lim_{x \to 0} \frac{\frac{x}{\sin (x)} \cdot \frac{x \cos (x) - \sin (x) \cdot 1}{x^{2}}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.8

Multiply $- 1$ by $1$ .

$e^{lim_{x \to 0} \frac{\frac{x}{\sin (x)} \cdot \frac{x \cos (x) - \sin (x)}{x^{2}}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.9

Multiply $\frac{x}{\sin (x)}$ by $\frac{x \cos (x) - \sin (x)}{x^{2}}$ .

$e^{lim_{x \to 0} \frac{\frac{x (x \cos (x) - \sin (x))}{\sin (x) x^{2}}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.10

Cancel the common factors.

Tap for more steps...

Step 3.3.10.1

Factor $x$ out of $\sin (x) x^{2}$ .

$e^{lim_{x \to 0} \frac{\frac{x (x \cos (x) - \sin (x))}{x \sin (x) x}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.10.2

Cancel the common factor.

$e^{lim_{x \to 0} \frac{\frac{x (x \cos (x) - \sin (x))}{x \sin (x) x}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.10.3

Rewrite the expression.

$e^{lim_{x \to 0} \frac{\frac{x \cos (x) - \sin (x)}{\sin (x) x}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.11

Reorder terms.

$e^{lim_{x \to 0} \frac{\frac{x \cos (x) - \sin (x)}{x \sin (x)}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{lim_{x \to 0} \frac{\frac{x \cos (x) - \sin (x)}{x \sin (x)}}{2 x}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x \sin (x)} \cdot \frac{1}{2 x}}$

Step 3.5

Combine factors.

Tap for more steps...

Step 3.5.1

Multiply $\frac{x \cos (x) - \sin (x)}{x \sin (x)}$ by $\frac{1}{2 x}$ .

$e^{lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x \sin (x) \cdot 2 x}}$

Step 3.5.2

Raise $x$ to the power of $1$ .

$e^{lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x^{1} x \sin (x) \cdot 2}}$

Step 3.5.3

Raise $x$ to the power of $1$ .

$e^{lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x^{1} x^{1} \sin (x) \cdot 2}}$

Step 3.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x^{1 + 1} \sin (x) \cdot 2}}$

Step 3.5.5

Add $1$ and $1$ .

$e^{lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x^{2} \sin (x) \cdot 2}}$

Step 4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x^{2} \sin (x)}}$

Step 5

Apply L'Hospital's rule.

Tap for more steps...

Step 5.1

Evaluate the limit of the numerator and the limit of the denominator.

Tap for more steps...

Step 5.1.1

Take the limit of the numerator and the limit of the denominator.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cos (x) - \sin (x)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 5.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cos (x) - lim_{x \to 0} \sin (x)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot lim_{x \to 0} \cos (x) - lim_{x \to 0} \sin (x)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.3

Move the limit inside the trig function because cosine is continuous.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) - lim_{x \to 0} \sin (x)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.4

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) - \sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 5.1.2.5.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \cos (lim_{x \to 0} x) - \sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.5.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \cos (0) - \sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.5.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \cos (0) - \sin (0)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.6

Simplify the answer.

Tap for more steps...

Step 5.1.2.6.1

Simplify each term.

Tap for more steps...

Step 5.1.2.6.1.1

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot 1 - \sin (0)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.6.1.2

Multiply $0$ by $1$ .

$e^{\frac{1}{2} \cdot \frac{0 - \sin (0)}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.6.1.3

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0 - 0}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.6.1.4

Multiply $- 1$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0 + 0}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.2.6.2

Add $0$ and $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x^{2} \sin (x)}}$

Step 5.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 5.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x^{2} \cdot lim_{x \to 0} \sin (x)}}$

Step 5.1.3.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$e^{\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} \sin (x)}}$

Step 5.1.3.3

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x)}}$

Step 5.1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 5.1.3.4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{0^{2} \cdot \sin (lim_{x \to 0} x)}}$

Step 5.1.3.4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{0^{2} \cdot \sin (0)}}$

Step 5.1.3.5

Simplify the answer.

Tap for more steps...

Step 5.1.3.5.1

Raising $0$ to any positive power yields $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 \cdot \sin (0)}}$

Step 5.1.3.5.2

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 \cdot 0}}$

Step 5.1.3.5.3

Multiply $0$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 5.1.3.5.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 5.1.3.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 5.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 5.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{x \cos (x) - \sin (x)}{x^{2} \sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x \cos (x) - \sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}$

Step 5.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 5.3.1

Differentiate the numerator and denominator.

$e^{\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [x \cos (x) - \sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.2

By the Sum Rule, the derivative of $x \cos (x) - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [x \cos (x)] + \frac{d}{d x} [- \sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [x \cos (x)] + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.3

Evaluate $\frac{d}{d x} [x \cos (x)]$ .

Tap for more steps...

Step 5.3.3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x] + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x) \frac{d}{d x} [x] + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x) \cdot 1 + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.3.4

Multiply $\cos (x)$ by $1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x) + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.4

Evaluate $\frac{d}{d x} [- \sin (x)]$ .

Tap for more steps...

Step 5.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x) - \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.4.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x) - \cos (x)}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.5

Simplify.

Tap for more steps...

Step 5.3.5.1

Combine terms.

Tap for more steps...

Step 5.3.5.1.1

Subtract $\cos (x)$ from $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + 0}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.5.1.2

Add $x (- \sin (x))$ and $0$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \cdot - 1 \sin (x)}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.5.2

Reorder the factors of $x (- \sin (x))$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \sin (x)}{\frac{d}{d x} [x^{2} \sin (x)]}}$

Step 5.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \sin (x)}{x^{2} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{2}]}}$

Step 5.3.7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \sin (x)}{x^{2} \cos (x) + \sin (x) \frac{d}{d x} [x^{2}]}}$

Step 5.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \sin (x)}{x^{2} \cos (x) + \sin (x) \cdot 2 x}}$

Step 5.3.9

Reorder terms.

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \sin (x)}{x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6

Apply L'Hospital's rule.

Tap for more steps...

Step 6.1

Evaluate the limit of the numerator and the limit of the denominator.

Tap for more steps...

Step 6.1.1

Take the limit of the numerator and the limit of the denominator.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} - x \sin (x)}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 6.1.2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} \cdot \frac{- lim_{x \to 0} x \sin (x)}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{- lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x)}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.2.3

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.2.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1.2.4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (lim_{x \to 0} x)}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.2.4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0)}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.2.5

Simplify the answer.

Tap for more steps...

Step 6.1.2.5.1

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot 0}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.2.5.2

Multiply $0$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x^{2} \cos (x) + 2 x \sin (x)}}$

Step 6.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 6.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x^{2} \cos (x) + lim_{x \to 0} 2 x \sin (x)}}$

Step 6.1.3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x^{2} \cdot lim_{x \to 0} \cos (x) + lim_{x \to 0} 2 x \sin (x)}}$

Step 6.1.3.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$e^{\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} \cos (x) + lim_{x \to 0} 2 x \sin (x)}}$

Step 6.1.3.4

Move the limit inside the trig function because cosine is continuous.

$e^{\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) + lim_{x \to 0} 2 x \sin (x)}}$

Step 6.1.3.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) + 2 lim_{x \to 0} x \sin (x)}}$

Step 6.1.3.6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) + 2 lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x)}}$

Step 6.1.3.7

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) + 2 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x)}}$

Step 6.1.3.8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1.3.8.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{0^{2} \cdot \cos (lim_{x \to 0} x) + 2 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x)}}$

Step 6.1.3.8.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{0^{2} \cdot \cos (0) + 2 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x)}}$

Step 6.1.3.8.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{0^{2} \cdot \cos (0) + 2 \cdot 0 \cdot \sin (lim_{x \to 0} x)}}$

Step 6.1.3.8.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{0^{2} \cdot \cos (0) + 2 \cdot 0 \cdot \sin (0)}}$

Step 6.1.3.9

Simplify the answer.

Tap for more steps...

Step 6.1.3.9.1

Simplify each term.

Tap for more steps...

Step 6.1.3.9.1.1

Raising $0$ to any positive power yields $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 \cdot \cos (0) + 2 \cdot 0 \cdot \sin (0)}}$

Step 6.1.3.9.1.2

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 \cdot 1 + 2 \cdot 0 \cdot \sin (0)}}$

Step 6.1.3.9.1.3

Multiply $0$ by $1$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 2 \cdot 0 \cdot \sin (0)}}$

Step 6.1.3.9.1.4

Multiply $2$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0 \cdot \sin (0)}}$

Step 6.1.3.9.1.5

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0 \cdot 0}}$

Step 6.1.3.9.1.6

Multiply $0$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0}}$

Step 6.1.3.9.2

Add $0$ and $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 6.1.3.9.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 6.1.3.10

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 6.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 6.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{- x \sin (x)}{x^{2} \cos (x) + 2 x \sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [- x \sin (x)]}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}$

Step 6.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 6.3.1

Differentiate the numerator and denominator.

$e^{\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [- x \sin (x)]}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}}$

Step 6.3.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [x \sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- \frac{d}{d x} [x \sin (x)]}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}}$

Step 6.3.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x])}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}}$

Step 6.3.4

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cos (x) + \sin (x) \frac{d}{d x} [x])}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}}$

Step 6.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cos (x) + \sin (x) \cdot 1)}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}}$

Step 6.3.6

Multiply $\sin (x)$ by $1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cos (x) + \sin (x))}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}}$

Step 6.3.7

Apply the distributive property.

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{\frac{d}{d x} [x^{2} \cos (x) + 2 x \sin (x)]}}$

Step 6.3.8

By the Sum Rule, the derivative of $x^{2} \cos (x) + 2 x \sin (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2} \cos (x)] + \frac{d}{d x} [2 x \sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{\frac{d}{d x} [x^{2} \cos (x)] + \frac{d}{d x} [2 x \sin (x)]}}$

Step 6.3.9

Evaluate $\frac{d}{d x} [x^{2} \cos (x)]$ .

Tap for more steps...

Step 6.3.9.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x \sin (x)]}}$

Step 6.3.9.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x \sin (x)]}}$

Step 6.3.9.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \cdot 2 x + \frac{d}{d x} [2 x \sin (x)]}}$

Step 6.3.10

Evaluate $\frac{d}{d x} [2 x \sin (x)]$ .

Tap for more steps...

Step 6.3.10.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [x \sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \cdot 2 x + 2 \frac{d}{d x} [x \sin (x)]}}$

Step 6.3.10.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \cdot 2 x + 2 (x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x])}}$

Step 6.3.10.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \cdot 2 x + 2 (x \cos (x) + \sin (x) \frac{d}{d x} [x])}}$

Step 6.3.10.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \cdot 2 x + 2 (x \cos (x) + \sin (x) \cdot 1)}}$

Step 6.3.10.5

Multiply $\sin (x)$ by $1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \cdot 2 x + 2 (x \cos (x) + \sin (x))}}$

Step 6.3.11

Simplify.

Tap for more steps...

Step 6.3.11.1

Apply the distributive property.

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + \cos (x) \cdot 2 x + 2 x \cos (x) + 2 \sin (x)}}$

Step 6.3.11.2

Add $\cos (x) (2 x)$ and $2 x \cos (x)$ .

Tap for more steps...

Step 6.3.11.2.1

Move $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + 2 x \cos (x) + 2 x \cos (x) + 2 \sin (x)}}$

Step 6.3.11.2.2

Add $2 x \cos (x)$ and $2 x \cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{x^{2} \cdot - 1 \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 6.3.11.3

Reorder terms.

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7

Apply L'Hospital's rule.

Tap for more steps...

Step 7.1

Evaluate the limit of the numerator and the limit of the denominator.

Tap for more steps...

Step 7.1.1

Take the limit of the numerator and the limit of the denominator.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} - x \cos (x) - \sin (x)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 7.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{- lim_{x \to 0} x \cos (x) - lim_{x \to 0} \sin (x)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{- lim_{x \to 0} x \cdot lim_{x \to 0} \cos (x) - lim_{x \to 0} \sin (x)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.3

Move the limit inside the trig function because cosine is continuous.

$e^{\frac{1}{2} \cdot \frac{- lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) - lim_{x \to 0} \sin (x)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.4

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{- lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) - \sin (lim_{x \to 0} x)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 7.1.2.5.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \cos (lim_{x \to 0} x) - \sin (lim_{x \to 0} x)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.5.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \cos (0) - \sin (lim_{x \to 0} x)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.5.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \cos (0) - \sin (0)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.6

Simplify the answer.

Tap for more steps...

Step 7.1.2.6.1

Simplify each term.

Tap for more steps...

Step 7.1.2.6.1.1

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot 1 - \sin (0)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.6.1.2

Multiply $0$ by $1$ .

$e^{\frac{1}{2} \cdot \frac{0 - \sin (0)}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.6.1.3

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0 - 0}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.6.1.4

Multiply $- 1$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0 + 0}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.2.6.2

Add $0$ and $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} - x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)}}$

Step 7.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 7.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{- lim_{x \to 0} x^{2} \sin (x) + lim_{x \to 0} 4 x \cos (x) + lim_{x \to 0} 2 \sin (x)}}$

Step 7.1.3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{- lim_{x \to 0} x^{2} \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} 4 x \cos (x) + lim_{x \to 0} 2 \sin (x)}}$

Step 7.1.3.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$e^{\frac{1}{2} \cdot \frac{0}{- {(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} 4 x \cos (x) + lim_{x \to 0} 2 \sin (x)}}$

Step 7.1.3.4

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{0}{- {(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x) + lim_{x \to 0} 4 x \cos (x) + lim_{x \to 0} 2 \sin (x)}}$

Step 7.1.3.5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{- {(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x) + 4 lim_{x \to 0} x \cos (x) + lim_{x \to 0} 2 \sin (x)}}$

Step 7.1.3.6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{- {(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x) + 4 lim_{x \to 0} x \cdot lim_{x \to 0} \cos (x) + lim_{x \to 0} 2 \sin (x)}}$

Step 7.1.3.7

Move the limit inside the trig function because cosine is continuous.

$e^{\frac{1}{2} \cdot \frac{0}{- {(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x) + 4 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) + lim_{x \to 0} 2 \sin (x)}}$

Step 7.1.3.8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{- {(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x) + 4 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) + 2 lim_{x \to 0} \sin (x)}}$

Step 7.1.3.9

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{0}{- {(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x) + 4 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) + 2 \sin (lim_{x \to 0} x)}}$

Step 7.1.3.10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 7.1.3.10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{- 0^{2} \cdot \sin (lim_{x \to 0} x) + 4 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) + 2 \sin (lim_{x \to 0} x)}}$

Step 7.1.3.10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{- 0^{2} \cdot \sin (0) + 4 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) + 2 \sin (lim_{x \to 0} x)}}$

Step 7.1.3.10.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{- 0^{2} \cdot \sin (0) + 4 \cdot 0 \cdot \cos (lim_{x \to 0} x) + 2 \sin (lim_{x \to 0} x)}}$

Step 7.1.3.10.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{- 0^{2} \cdot \sin (0) + 4 \cdot 0 \cdot \cos (0) + 2 \sin (lim_{x \to 0} x)}}$

Step 7.1.3.10.5

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0}{- 0^{2} \cdot \sin (0) + 4 \cdot 0 \cdot \cos (0) + 2 \sin (0)}}$

Step 7.1.3.11

Simplify the answer.

Tap for more steps...

Step 7.1.3.11.1

Simplify each term.

Tap for more steps...

Step 7.1.3.11.1.1

Raising $0$ to any positive power yields $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{- 0 \cdot \sin (0) + 4 \cdot 0 \cdot \cos (0) + 2 \sin (0)}}$

Step 7.1.3.11.1.2

Multiply $- 1$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 \cdot \sin (0) + 4 \cdot 0 \cdot \cos (0) + 2 \sin (0)}}$

Step 7.1.3.11.1.3

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 \cdot 0 + 4 \cdot 0 \cdot \cos (0) + 2 \sin (0)}}$

Step 7.1.3.11.1.4

Multiply $0$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 4 \cdot 0 \cdot \cos (0) + 2 \sin (0)}}$

Step 7.1.3.11.1.5

Multiply $4$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0 \cdot \cos (0) + 2 \sin (0)}}$

Step 7.1.3.11.1.6

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0 \cdot 1 + 2 \sin (0)}}$

Step 7.1.3.11.1.7

Multiply $0$ by $1$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0 + 2 \sin (0)}}$

Step 7.1.3.11.1.8

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0 + 2 \cdot 0}}$

Step 7.1.3.11.1.9

Multiply $2$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0 + 0}}$

Step 7.1.3.11.2

Add $0$ and $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0 + 0}}$

Step 7.1.3.11.3

Add $0$ and $0$ .

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 7.1.3.11.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 7.1.3.12

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 7.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{1}{2} \cdot \frac{0}{0}}$

Step 7.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{- x \cos (x) - \sin (x)}{- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [- x \cos (x) - \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}$

Step 7.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 7.3.1

Differentiate the numerator and denominator.

$e^{\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [- x \cos (x) - \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.2

By the Sum Rule, the derivative of $- x \cos (x) - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [- x \cos (x)] + \frac{d}{d x} [- \sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [- x \cos (x)] + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.3

Evaluate $\frac{d}{d x} [- x \cos (x)]$ .

Tap for more steps...

Step 7.3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [x \cos (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- \frac{d}{d x} [x \cos (x)] + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x]) + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cdot - 1 \sin (x) + \cos (x) \frac{d}{d x} [x]) + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cdot - 1 \sin (x) + \cos (x) \cdot 1) + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.3.5

Multiply $\cos (x)$ by $1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cdot - 1 \sin (x) + \cos (x)) + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.4

Evaluate $\frac{d}{d x} [- \sin (x)]$ .

Tap for more steps...

Step 7.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cdot - 1 \sin (x) + \cos (x)) - \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.4.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{- (x \cdot - 1 \sin (x) + \cos (x)) - \cos (x)}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.5

Simplify.

Tap for more steps...

Step 7.3.5.1

Apply the distributive property.

$e^{\frac{1}{2} lim_{x \to 0} \frac{- x \cdot - 1 \sin (x) - \cos (x) - \cos (x)}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.5.2

Combine terms.

Tap for more steps...

Step 7.3.5.2.1

Multiply $- 1$ by $- 1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{1 x \sin (x) - \cos (x) - \cos (x)}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.5.2.2

Multiply $x$ by $1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - \cos (x) - \cos (x)}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.5.2.3

Subtract $\cos (x)$ from $- \cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{\frac{d}{d x} [- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)]}}$

Step 7.3.6

By the Sum Rule, the derivative of $- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [- x^{2} \sin (x)] + \frac{d}{d x} [4 x \cos (x)] + \frac{d}{d x} [2 \sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{\frac{d}{d x} [- x^{2} \sin (x)] + \frac{d}{d x} [4 x \cos (x)] + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.7

Evaluate $\frac{d}{d x} [- x^{2} \sin (x)]$ .

Tap for more steps...

Step 7.3.7.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2} \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [x^{2} \sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- \frac{d}{d x} [x^{2} \sin (x)] + \frac{d}{d x} [4 x \cos (x)] + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.7.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [4 x \cos (x)] + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.7.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [4 x \cos (x)] + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.7.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + \frac{d}{d x} [4 x \cos (x)] + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.8

Evaluate $\frac{d}{d x} [4 x \cos (x)]$ .

Tap for more steps...

Step 7.3.8.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x \cos (x)$ with respect to $x$ is $4 \frac{d}{d x} [x \cos (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + 4 \frac{d}{d x} [x \cos (x)] + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.8.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + 4 (x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x]) + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.8.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + 4 (x \cdot - 1 \sin (x) + \cos (x) \frac{d}{d x} [x]) + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.8.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + 4 (x \cdot - 1 \sin (x) + \cos (x) \cdot 1) + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.8.5

Multiply $\cos (x)$ by $1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + 4 (x \cdot - 1 \sin (x) + \cos (x)) + \frac{d}{d x} [2 \sin (x)]}}$

Step 7.3.9

Evaluate $\frac{d}{d x} [2 \sin (x)]$ .

Tap for more steps...

Step 7.3.9.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin (x)]$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + 4 (x \cdot - 1 \sin (x) + \cos (x)) + 2 \frac{d}{d x} [\sin (x)]}}$

Step 7.3.9.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- (x^{2} \cos (x) + \sin (x) \cdot 2 x) + 4 (x \cdot - 1 \sin (x) + \cos (x)) + 2 \cos (x)}}$

Step 7.3.10

Simplify.

Tap for more steps...

Step 7.3.10.1

Apply the distributive property.

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- x^{2} \cos (x) - \sin (x) \cdot 2 x + 4 (x \cdot - 1 \sin (x) + \cos (x)) + 2 \cos (x)}}$

Step 7.3.10.2

Apply the distributive property.

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- x^{2} \cos (x) - \sin (x) \cdot 2 x + 4 x \cdot - 1 \sin (x) + 4 \cos (x) + 2 \cos (x)}}$

Step 7.3.10.3

Combine terms.

Tap for more steps...

Step 7.3.10.3.1

Multiply $2$ by $- 1$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- x^{2} \cos (x) - 2 \sin (x) x + 4 x \cdot - 1 \sin (x) + 4 \cos (x) + 2 \cos (x)}}$

Step 7.3.10.3.2

Multiply $- 1$ by $4$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- x^{2} \cos (x) - 2 \sin (x) x - 4 x \sin (x) + 4 \cos (x) + 2 \cos (x)}}$

Step 7.3.10.3.3

Subtract $4 x \sin (x)$ from $- 2 \sin (x) x$ .

Tap for more steps...

Step 7.3.10.3.3.1

Move $\sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- x^{2} \cos (x) - 2 x \sin (x) - 4 x \sin (x) + 4 \cos (x) + 2 \cos (x)}}$

Step 7.3.10.3.3.2

Subtract $4 x \sin (x)$ from $- 2 x \sin (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- x^{2} \cos (x) - 6 x \sin (x) + 4 \cos (x) + 2 \cos (x)}}$

Step 7.3.10.3.4

Add $4 \cos (x)$ and $2 \cos (x)$ .

$e^{\frac{1}{2} lim_{x \to 0} \frac{x \sin (x) - 2 \cos (x)}{- x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)}}$

Step 8

Evaluate the limit.

Tap for more steps...

Step 8.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \sin (x) - 2 \cos (x)}{lim_{x \to 0} - x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)}}$

Step 8.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \sin (x) - lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)}}$

Step 8.3

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)}}$

Step 8.4

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)}}$

Step 8.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 lim_{x \to 0} \cos (x)}{lim_{x \to 0} - x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)}}$

Step 8.6

Move the limit inside the trig function because cosine is continuous.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{lim_{x \to 0} - x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)}}$

Step 8.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- lim_{x \to 0} x^{2} \cos (x) - lim_{x \to 0} 6 x \sin (x) + lim_{x \to 0} 6 \cos (x)}}$

Step 8.8

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- lim_{x \to 0} x^{2} \cdot lim_{x \to 0} \cos (x) - lim_{x \to 0} 6 x \sin (x) + lim_{x \to 0} 6 \cos (x)}}$

Step 8.9

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- {(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} \cos (x) - lim_{x \to 0} 6 x \sin (x) + lim_{x \to 0} 6 \cos (x)}}$

Step 8.10

Move the limit inside the trig function because cosine is continuous.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- {(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) - lim_{x \to 0} 6 x \sin (x) + lim_{x \to 0} 6 \cos (x)}}$

Step 8.11

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- {(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) - 6 lim_{x \to 0} x \sin (x) + lim_{x \to 0} 6 \cos (x)}}$

Step 8.12

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- {(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} 6 \cos (x)}}$

Step 8.13

Move the limit inside the trig function because sine is continuous.

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- {(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + lim_{x \to 0} 6 \cos (x)}}$

Step 8.14

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

Step 8.15

Move the limit inside the trig function because cosine is continuous.

Step 9

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 9.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (lim_{x \to 0} x) - 2 \cos (lim_{x \to 0} x)}{- {(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x)}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0) - 2 \cos (lim_{x \to 0} x)}{- {(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x)}}$

Step 9.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0) - 2 \cos (0)}{- {(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x)}}$

Step 9.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0) - 2 \cos (0)}{- 0^{2} \cdot \cos (lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x)}}$

Step 9.5

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0) - 2 \cos (0)}{- 0^{2} \cdot \cos (0) - 6 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x)}}$

Step 9.6

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0) - 2 \cos (0)}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x)}}$

Step 9.7

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0) - 2 \cos (0)}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (lim_{x \to 0} x)}}$

Step 9.8

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot \sin (0) - 2 \cos (0)}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{0 \cdot 0 - 2 \cos (0)}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.1.2

Multiply $0$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{0 - 2 \cos (0)}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.1.3

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{1}{2} \cdot \frac{0 - 2 \cdot 1}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.1.4

Multiply $- 2$ by $1$ .

$e^{\frac{1}{2} \cdot \frac{0 - 2}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.1.5

Subtract $2$ from $0$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{- 0^{2} \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

Raising $0$ to any positive power yields $0$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{- 0 \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.2.2

Multiply $- 1$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 \cdot \cos (0) - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.2.3

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 \cdot 1 - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.2.4

Multiply $0$ by $1$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 - 6 \cdot 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.2.5

Multiply $- 6$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 + 0 \cdot \sin (0) + 6 \cos (0)}}$

Step 10.2.6

The exact value of $\sin (0)$ is $0$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 + 0 \cdot 0 + 6 \cos (0)}}$

Step 10.2.7

Multiply $0$ by $0$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 + 0 + 6 \cos (0)}}$

Step 10.2.8

The exact value of $\cos (0)$ is $1$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 + 0 + 6 \cdot 1}}$

Step 10.2.9

Multiply $6$ by $1$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 + 0 + 6}}$

Step 10.2.10

Add $0$ and $0$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{0 + 6}}$

Step 10.2.11

Add $0$ and $6$ .

$e^{\frac{1}{2} \cdot \frac{- 2}{6}}$

Step 10.3

Cancel the common factor of $- 2$ and $6$ .

Tap for more steps...

Step 10.3.1

Factor $2$ out of $- 2$ .

$e^{\frac{1}{2} \cdot \frac{2 (- 1)}{6}}$

Step 10.3.2

Cancel the common factors.

Tap for more steps...

Step 10.3.2.1

Factor $2$ out of $6$ .

$e^{\frac{1}{2} \cdot \frac{2 \cdot - 1}{2 \cdot 3}}$

Step 10.3.2.2

Cancel the common factor.

$e^{\frac{1}{2} \cdot \frac{2 \cdot - 1}{2 \cdot 3}}$

Step 10.3.2.3

Rewrite the expression.

$e^{\frac{1}{2} \cdot \frac{- 1}{3}}$

Step 10.4

Move the negative in front of the fraction.

$e^{\frac{1}{2} (- \frac{1}{3})}$

Step 10.5

Multiply $\frac{1}{2} (- \frac{1}{3})$ .

Tap for more steps...

Step 10.5.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$e^{- \frac{1}{2 \cdot 3}}$

Step 10.5.2

Multiply $2$ by $3$ .

$e^{- \frac{1}{6}}$

Step 10.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{e^{\frac{1}{6}}}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e^{\frac{1}{6}}}$

Decimal Form:

$0.84648172 \dots$