Calculus Examples

$lim_{x \to 0} \frac{\sin (x) (1 - \cos (x))}{3 x^{2}}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) (1 - \cos (x))}{x^{2}}$

Step 2

Apply L'Hospital's rule.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} \sin (x) (1 - \cos (x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 2.1.2.1

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} 1 - \cos (x)}{lim_{x \to 0} x^{2}}$

Step 2.1.2.2

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

$\frac{1}{3} \cdot \frac{\sin (lim_{x \to 0} x) \cdot lim_{x \to 0} 1 - \cos (x)}{lim_{x \to 0} x^{2}}$

Step 2.1.2.3

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

$\frac{1}{3} \cdot \frac{\sin (lim_{x \to 0} x) \cdot (lim_{x \to 0} 1 - lim_{x \to 0} \cos (x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} \cdot \frac{\sin (lim_{x \to 0} x) \cdot (1 - lim_{x \to 0} \cos (x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.5

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

$\frac{1}{3} \cdot \frac{\sin (lim_{x \to 0} x) \cdot (1 - \cos (lim_{x \to 0} x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.6

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

Tap for more steps...

Step 2.1.2.6.1

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

$\frac{1}{3} \cdot \frac{\sin (0) \cdot (1 - \cos (lim_{x \to 0} x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.6.2

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

$\frac{1}{3} \cdot \frac{\sin (0) \cdot (1 - \cos (0))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.7

Simplify the answer.

Tap for more steps...

Step 2.1.2.7.1

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

$\frac{1}{3} \cdot \frac{0 \cdot (1 - \cos (0))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.7.2

Simplify each term.

Tap for more steps...

Step 2.1.2.7.2.1

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

$\frac{1}{3} \cdot \frac{0 \cdot (1 - 1 \cdot 1)}{lim_{x \to 0} x^{2}}$

Step 2.1.2.7.2.2

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{0 \cdot (1 - 1)}{lim_{x \to 0} x^{2}}$

Step 2.1.2.7.3

Subtract $1$ from $1$ .

$\frac{1}{3} \cdot \frac{0 \cdot 0}{lim_{x \to 0} x^{2}}$

Step 2.1.2.7.4

Multiply $0$ by $0$ .

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x^{2}}$

Step 2.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 2.1.3.1

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

$\frac{1}{3} \cdot \frac{0}{{(lim_{x \to 0} x)}^{2}}$

Step 2.1.3.2

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

$\frac{1}{3} \cdot \frac{0}{0^{2}}$

Step 2.1.3.3

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

$\frac{1}{3} \cdot \frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x) (1 - \cos (x))]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.2

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 $f (x) = \sin (x)$ and $g (x) = 1 - \cos (x)$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) \frac{d}{d x} [1 - \cos (x)] + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.3

By the Sum Rule, the derivative of $1 - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) (\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)]) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) (0 + \frac{d}{d x} [- \cos (x)]) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.5

Add $0$ and $\frac{d}{d x} [- \cos (x)]$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) \frac{d}{d x} [- \cos (x)] + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.6

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

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) \cdot - 1 \frac{d}{d x} [\cos (x)] + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.7

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

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) \cdot - 1 \cdot - 1 \sin (x) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.8

Multiply $- 1$ by $- 1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) \cdot 1 \sin (x) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.9

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

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x) \sin (x) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.10

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{1} (x) \sin (x) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.11

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{1} (x) \sin^{1} (x) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.12

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

$\frac{1}{3} lim_{x \to 0} \frac{{\sin (x)}^{1 + 1} + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.13

Add $1$ and $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + (1 - \cos (x)) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.14

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

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + (1 - \cos (x)) \cos (x)}{\frac{d}{d x} [x^{2}]}$

Step 2.3.15

Simplify.

Tap for more steps...

Step 2.3.15.1

Apply the distributive property.

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + 1 \cos (x) - \cos (x) \cos (x)}{\frac{d}{d x} [x^{2}]}$

Step 2.3.15.2

Combine terms.

Tap for more steps...

Step 2.3.15.2.1

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

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - \cos (x) \cos (x)}{\frac{d}{d x} [x^{2}]}$

Step 2.3.15.2.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - \cos^{1} (x) \cos (x)}{\frac{d}{d x} [x^{2}]}$

Step 2.3.15.2.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - \cos^{1} (x) \cos^{1} (x)}{\frac{d}{d x} [x^{2}]}$

Step 2.3.15.2.4

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

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - {\cos (x)}^{1 + 1}}{\frac{d}{d x} [x^{2}]}$

Step 2.3.15.2.5

Add $1$ and $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - \cos^{2} (x)}{\frac{d}{d x} [x^{2}]}$

Step 2.3.16

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

$\frac{1}{3} lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - \cos^{2} (x)}{2 x}$

Step 3

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{\sin^{2} (x) + \cos (x) - \cos^{2} (x)}{x}$

Step 4

Apply L'Hospital's rule.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{lim_{x \to 0} \sin^{2} (x) + \cos (x) - \cos^{2} (x)}{lim_{x \to 0} x}$

Step 4.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 4.1.2.1

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{lim_{x \to 0} \sin^{2} (x) + lim_{x \to 0} \cos (x) - lim_{x \to 0} \cos^{2} (x)}{lim_{x \to 0} x}$

Step 4.1.2.2

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{{(lim_{x \to 0} \sin (x))}^{2} + lim_{x \to 0} \cos (x) - lim_{x \to 0} \cos^{2} (x)}{lim_{x \to 0} x}$

Step 4.1.2.3

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{\sin^{2} (lim_{x \to 0} x) + lim_{x \to 0} \cos (x) - lim_{x \to 0} \cos^{2} (x)}{lim_{x \to 0} x}$

Step 4.1.2.4

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{\sin^{2} (lim_{x \to 0} x) + \cos (lim_{x \to 0} x) - lim_{x \to 0} \cos^{2} (x)}{lim_{x \to 0} x}$

Step 4.1.2.5

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{\sin^{2} (lim_{x \to 0} x) + \cos (lim_{x \to 0} x) - {(lim_{x \to 0} \cos (x))}^{2}}{lim_{x \to 0} x}$

Step 4.1.2.6

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{\sin^{2} (lim_{x \to 0} x) + \cos (lim_{x \to 0} x) - \cos^{2} (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 4.1.2.7

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

Tap for more steps...

Step 4.1.2.7.1

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{\sin^{2} (0) + \cos (lim_{x \to 0} x) - \cos^{2} (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 4.1.2.7.2

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{\sin^{2} (0) + \cos (0) - \cos^{2} (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 4.1.2.7.3

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{\sin^{2} (0) + \cos (0) - \cos^{2} (0)}{lim_{x \to 0} x}$

Step 4.1.2.8

Simplify the answer.

Tap for more steps...

Step 4.1.2.8.1

Simplify each term.

Tap for more steps...

Step 4.1.2.8.1.1

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{0^{2} + \cos (0) - \cos^{2} (0)}{lim_{x \to 0} x}$

Step 4.1.2.8.1.2

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{0 + \cos (0) - \cos^{2} (0)}{lim_{x \to 0} x}$

Step 4.1.2.8.1.3

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{0 + 1 - \cos^{2} (0)}{lim_{x \to 0} x}$

Step 4.1.2.8.1.4

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{0 + 1 - 1^{2}}{lim_{x \to 0} x}$

Step 4.1.2.8.1.5

One to any power is one.

$\frac{1}{3} \cdot \frac{1}{2} \frac{0 + 1 - 1 \cdot 1}{lim_{x \to 0} x}$

Step 4.1.2.8.1.6

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{1}{2} \frac{0 + 1 - 1}{lim_{x \to 0} x}$

Step 4.1.2.8.2

Add $0$ and $1$ .

$\frac{1}{3} \cdot \frac{1}{2} \frac{1 - 1}{lim_{x \to 0} x}$

Step 4.1.2.8.3

Subtract $1$ from $1$ .

$\frac{1}{3} \cdot \frac{1}{2} \frac{0}{lim_{x \to 0} x}$

Step 4.1.3

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

$\frac{1}{3} \cdot \frac{1}{2} \frac{0}{0}$

Step 4.1.4

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

Undefined

$\frac{1}{3} \cdot \frac{1}{2} \frac{0}{0}$

Step 4.2

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 4.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [\sin^{2} (x) + \cos (x) - \cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.2

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [\sin^{2} (x)] + \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.3

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

Tap for more steps...

Step 4.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

Tap for more steps...

Step 4.3.3.1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x)$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.3.1.2

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 u_{1} \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.3.1.3

Replace all occurrences of $u_{1}$ with $\sin (x)$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.3.2

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) + \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.4

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) + \frac{d}{d x} [- \cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.5

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

Tap for more steps...

Step 4.3.5.1

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) - \frac{d}{d x} [\cos^{2} (x)]}{\frac{d}{d x} [x]}$

Step 4.3.5.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

Tap for more steps...

Step 4.3.5.2.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) - \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x]}$

Step 4.3.5.2.2

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) - 1 \cdot 2 u_{2} \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x]}$

Step 4.3.5.2.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) - 1 \cdot 2 \cos (x) \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x]}$

Step 4.3.5.3

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) - 1 \cdot 2 \cos (x) \cdot - 1 \sin (x)}{\frac{d}{d x} [x]}$

Step 4.3.5.4

Multiply $- 1$ by $2$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) - - 2 \cos (x) \sin (x)}{\frac{d}{d x} [x]}$

Step 4.3.5.5

Multiply $- 2$ by $- 1$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \sin (x) \cos (x) - \sin (x) + 2 \cos (x) \sin (x)}{\frac{d}{d x} [x]}$

Step 4.3.6

Combine terms.

Tap for more steps...

Step 4.3.6.1

Reorder the factors of $2 \sin (x) \cos (x)$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{2 \cos (x) \sin (x) - \sin (x) + 2 \cos (x) \sin (x)}{\frac{d}{d x} [x]}$

Step 4.3.6.2

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{4 \cos (x) \sin (x) - \sin (x)}{\frac{d}{d x} [x]}$

Step 4.3.7

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

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} \frac{4 \cos (x) \sin (x) - \sin (x)}{1}$

Step 4.4

Divide $4 \cos (x) \sin (x) - \sin (x)$ by $1$ .

$\frac{1}{3} \cdot \frac{1}{2} lim_{x \to 0} 4 \cos (x) \sin (x) - \sin (x)$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

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

$\frac{1}{3} \cdot \frac{1}{2} (lim_{x \to 0} 4 \cos (x) \sin (x) - lim_{x \to 0} \sin (x))$

Step 5.2

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

$\frac{1}{3} \cdot \frac{1}{2} (4 lim_{x \to 0} \cos (x) \sin (x) - lim_{x \to 0} \sin (x))$

Step 5.3

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

$\frac{1}{3} \cdot \frac{1}{2} (4 lim_{x \to 0} \cos (x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} \sin (x))$

Step 5.4

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

$\frac{1}{3} \cdot \frac{1}{2} (4 \cos (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} \sin (x))$

Step 5.5

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

$\frac{1}{3} \cdot \frac{1}{2} (4 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - lim_{x \to 0} \sin (x))$

Step 5.6

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

$\frac{1}{3} \cdot \frac{1}{2} (4 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - \sin (lim_{x \to 0} x))$

Step 6

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

Tap for more steps...

Step 6.1

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

$\frac{1}{3} \cdot \frac{1}{2} (4 \cos (0) \cdot \sin (lim_{x \to 0} x) - \sin (lim_{x \to 0} x))$

Step 6.2

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

$\frac{1}{3} \cdot \frac{1}{2} (4 \cos (0) \cdot \sin (0) - \sin (lim_{x \to 0} x))$

Step 6.3

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

$\frac{1}{3} \cdot \frac{1}{2} (4 \cos (0) \cdot \sin (0) - \sin (0))$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

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

$\frac{1}{3 \cdot 2} (4 \cos (0) \cdot \sin (0) - \sin (0))$

Step 7.1.2

Multiply $3$ by $2$ .

$\frac{1}{6} (4 \cos (0) \cdot \sin (0) - \sin (0))$

Step 7.2

Simplify each term.

Tap for more steps...

Step 7.2.1

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

$\frac{1}{6} (4 \cdot 1 \cdot \sin (0) - \sin (0))$

Step 7.2.2

Multiply $4$ by $1$ .

$\frac{1}{6} (4 \cdot \sin (0) - \sin (0))$

Step 7.2.3

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

$\frac{1}{6} (4 \cdot 0 - \sin (0))$

Step 7.2.4

Multiply $4$ by $0$ .

$\frac{1}{6} (0 - \sin (0))$

Step 7.2.5

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

$\frac{1}{6} (0 - 0)$

Step 7.2.6

Multiply $- 1$ by $0$ .

$\frac{1}{6} (0 + 0)$

Step 7.3

Add $0$ and $0$ .

$\frac{1}{6} \cdot 0$

Step 7.4

Multiply $\frac{1}{6}$ by $0$ .

$0$