Calculus Examples

$lim_{x \to 0} \frac{x - \sin (x)}{x - \tan (x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 0} x - \sin (x)}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 0} x - lim_{x \to 0} \sin (x)}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.2.2

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

$\frac{lim_{x \to 0} x - \sin (lim_{x \to 0} x)}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$\frac{0 - \sin (lim_{x \to 0} x)}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.2.3.2

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

$\frac{0 - \sin (0)}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.2.4

Simplify the answer.

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Step 1.1.2.4.1

Simplify each term.

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Step 1.1.2.4.1.1

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

$\frac{0 - 0}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.2.4.1.2

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.2.4.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} x - \tan (x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 0} x - lim_{x \to 0} \tan (x)}$

Step 1.1.3.2

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

$\frac{0}{lim_{x \to 0} x - \tan (lim_{x \to 0} x)}$

Step 1.1.3.3

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

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Step 1.1.3.3.1

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

$\frac{0}{0 - \tan (lim_{x \to 0} x)}$

Step 1.1.3.3.2

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

$\frac{0}{0 - \tan (0)}$

Step 1.1.3.4

Simplify the answer.

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Step 1.1.3.4.1

Simplify each term.

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Step 1.1.3.4.1.1

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

$\frac{0}{0 - 0}$

Step 1.1.3.4.1.2

Multiply $- 1$ by $0$ .

$\frac{0}{0 + 0}$

Step 1.1.3.4.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.4.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.5

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [x - \sin (x)]}{\frac{d}{d x} [x - \tan (x)]}$

Step 1.3.2

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

$lim_{x \to 0} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- \sin (x)]}{\frac{d}{d x} [x - \tan (x)]}$

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

$lim_{x \to 0} \frac{1 - \cos (x)}{\frac{d}{d x} [x - \tan (x)]}$

Step 1.3.5

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

$lim_{x \to 0} \frac{1 - \cos (x)}{\frac{d}{d x} [x] + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.6

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

$lim_{x \to 0} \frac{1 - \cos (x)}{1 + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.7

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

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Step 1.3.7.1

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

$lim_{x \to 0} \frac{1 - \cos (x)}{1 - \frac{d}{d x} [\tan (x)]}$

Step 1.3.7.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

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

Step 2.1.2.1.2

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

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

Step 2.1.2.1.3

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

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

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

Step 2.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.2.3.2

Subtract $1$ from $1$ .

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

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

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

Step 2.1.3.1.2

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

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

Step 2.1.3.1.3

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

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

Step 2.1.3.1.4

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

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

Step 2.1.3.2

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

$\frac{0}{1 - \sec^{2} (0)}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Reorder $1$ and $- \sec^{2} (0)$ .

$\frac{0}{- \sec^{2} (0) + 1}$

Step 2.1.3.3.2

Factor $- 1$ out of $- \sec^{2} (0)$ .

$\frac{0}{- (\sec^{2} (0)) + 1}$

Step 2.1.3.3.3

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{0}{- \sec^{2} (0) - 1 \cdot - 1}$

Step 2.1.3.3.4

Factor $- 1$ out of $- \sec^{2} (0) - 1 \cdot - 1$ .

$\frac{0}{- (\sec^{2} (0) - 1)}$

Step 2.1.3.3.5

Apply pythagorean identity.

$\frac{0}{- \tan^{2} (0)}$

Step 2.1.3.3.6

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

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

Step 2.1.3.3.7

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

$\frac{0}{- 0}$

Step 2.1.3.3.8

Multiply $- 1$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.3.9

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

Undefined

$\frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4.4

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

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

Step 2.3.5

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

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

Step 2.3.6

By the Sum Rule, the derivative of $1 - \sec^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \sec^{2} (x)]$ .

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

Step 2.3.7

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

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

Step 2.3.8

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

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Step 2.3.8.1

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

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

Step 2.3.8.2

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 $g (x) = \sec (x)$ .

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Step 2.3.8.2.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 2.3.8.2.2

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

$lim_{x \to 0} \frac{\sin (x)}{0 - (2 u \frac{d}{d x} [\sec (x)])}$

Step 2.3.8.2.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 2.3.8.3

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$lim_{x \to 0} \frac{\sin (x)}{0 - (2 \sec (x) (\sec (x) \tan (x)))}$

Step 2.3.8.4

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

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

Step 2.3.8.5

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

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

Step 2.3.8.6

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

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

Step 2.3.8.7

Add $1$ and $1$ .

$lim_{x \to 0} \frac{\sin (x)}{0 - (2 \sec^{2} (x) \tan (x))}$

Step 2.3.8.8

Multiply $2$ by $- 1$ .

$lim_{x \to 0} \frac{\sin (x)}{0 - 2 \sec^{2} (x) \tan (x)}$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Subtract $2 \sec^{2} (x) \tan (x)$ from $0$ .

$lim_{x \to 0} \frac{\sin (x)}{- 2 \sec^{2} (x) \tan (x)}$

Step 2.3.9.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 2.3.9.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 2.3.9.4

One to any power is one.

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

Step 2.3.9.5

Combine $- 2$ and $\frac{1}{\cos^{2} (x)}$ .

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

Step 2.3.9.6

Move the negative in front of the fraction.

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

Step 2.3.9.7

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.3.9.8

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

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Step 2.3.9.8.1

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

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

Step 2.3.9.8.2

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 2.3.9.8.2.1

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

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Step 2.3.9.8.2.1.1

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

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

Step 2.3.9.8.2.1.2

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

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

Step 2.3.9.8.2.2

Add $1$ and $2$ .

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

Step 2.3.9.9

Move $2$ to the left of $\sin (x)$ .

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

Combine $\sin (x)$ and $\frac{\cos^{3} (x)}{2 \sin (x)}$ .

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

Step 2.6

Cancel the common factor of $\sin (x)$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Move the exponent $3$ from $\cos^{3} (x)$ outside the limit using the Limits Power Rule.

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

Step 3.4

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

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

Step 4

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

$- \frac{1}{2} \cos^{3} (0)$

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

One to any power is one.

$- \frac{1}{2} \cdot 1$

Step 5.3

Multiply $- 1$ by $1$ .

$- \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$