Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.1.2.2

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

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

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 $x$ .

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

Step 1.1.2.3.2

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

$\frac{\tan (0) + 0}{lim_{x \to 0} x^{3}}$

Step 1.1.2.4

Simplify the answer.

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Step 1.1.2.4.1

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

$\frac{0 + 0}{lim_{x \to 0} x^{3}}$

Step 1.1.2.4.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} x^{3}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

$\frac{0}{0^{3}}$

Step 1.1.3.3

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

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 0} \frac{\tan (x) - x}{x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} [\tan (x) - x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\tan (x) - x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3.2

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

$lim_{x \to 0} \frac{\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [- x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $- 1$ by $1$ .

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

Step 1.3.5

Reorder terms.

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

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

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

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

Step 3.1.2.1.4

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

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

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

Step 3.1.2.3.2

Apply pythagorean identity.

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

Step 3.1.2.3.3

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

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

Step 3.1.2.3.4

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

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

Undefined

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

Step 3.1.4

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

Undefined

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

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

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

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Step 3.3.4.1.1

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

$\frac{1}{3} lim_{x \to 0} \frac{0 + \frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)]}{\frac{d}{d x} [x^{2}]}$

Step 3.3.4.1.2

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

$\frac{1}{3} lim_{x \to 0} \frac{0 + 2 u \frac{d}{d x} [\sec (x)]}{\frac{d}{d x} [x^{2}]}$

Step 3.3.4.1.3

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

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

Step 3.3.4.2

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

$\frac{1}{3} lim_{x \to 0} \frac{0 + 2 \sec (x) \sec (x) \tan (x)}{\frac{d}{d x} [x^{2}]}$

Step 3.3.4.3

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

$\frac{1}{3} lim_{x \to 0} \frac{0 + 2 \sec^{1} (x) \sec (x) \tan (x)}{\frac{d}{d x} [x^{2}]}$

Step 3.3.4.4

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

$\frac{1}{3} lim_{x \to 0} \frac{0 + 2 \sec^{1} (x) \sec^{1} (x) \tan (x)}{\frac{d}{d x} [x^{2}]}$

Step 3.3.4.5

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

$\frac{1}{3} lim_{x \to 0} \frac{0 + 2 {\sec (x)}^{1 + 1} \tan (x)}{\frac{d}{d x} [x^{2}]}$

Step 3.3.4.6

Add $1$ and $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 + 2 \sec^{2} (x) \tan (x)}{\frac{d}{d x} [x^{2}]}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Add $0$ and $2 \sec^{2} (x) \tan (x)$ .

$\frac{1}{3} lim_{x \to 0} \frac{2 \sec^{2} (x) \tan (x)}{\frac{d}{d x} [x^{2}]}$

Step 3.3.5.2

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

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

Step 3.3.5.3

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

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

Step 3.3.5.4

One to any power is one.

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

Step 3.3.5.5

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

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

Step 3.3.5.6

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

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

Step 3.3.5.7

Combine.

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

Step 3.3.5.8

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

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Step 3.3.5.8.1

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

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Step 3.3.5.8.1.1

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

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

Step 3.3.5.8.1.2

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

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

Step 3.3.5.8.2

Add $2$ and $1$ .

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

Step 3.3.6

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Multiply $\frac{2 \sin (x)}{\cos^{3} (x)}$ by $\frac{1}{2 x}$ .

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

Step 3.6

Cancel the common factor of $2$ .

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

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Step 4.1.3.4.1

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

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

Step 4.1.3.4.2

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

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

Step 4.1.3.5

Simplify the answer.

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Step 4.1.3.5.1

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

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

Step 4.1.3.5.2

One to any power is one.

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

Step 4.1.3.5.3

Multiply $0$ by $1$ .

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

Step 4.1.3.5.4

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

Undefined

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

Step 4.1.3.6

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

Undefined

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

Step 4.1.4

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

Undefined

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

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

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

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

Step 4.3.3

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 $f (x) = \cos^{3} (x)$ and $g (x) = x$ .

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

Step 4.3.4

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

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

Step 4.3.5

Multiply $\cos^{3} (x)$ by $1$ .

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

Step 4.3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (x)$ .

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Step 4.3.6.1

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

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

Step 4.3.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

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

Step 4.3.6.3

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

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

Step 4.3.7

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

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

Step 4.3.8

Multiply $- 1$ by $3$ .

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

Step 4.3.9

Reorder terms.

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $- 3$ by $0$ .

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

Step 7.2.2

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

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

Step 7.2.3

One to any power is one.

$\frac{1}{3} \cdot \frac{1}{0 \cdot 1 \cdot \sin (0) + \cos^{3} (0)}$

Step 7.2.4

Multiply $0$ by $1$ .

$\frac{1}{3} \cdot \frac{1}{0 \cdot \sin (0) + \cos^{3} (0)}$

Step 7.2.5

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

$\frac{1}{3} \cdot \frac{1}{0 \cdot 0 + \cos^{3} (0)}$

Step 7.2.6

Multiply $0$ by $0$ .

$\frac{1}{3} \cdot \frac{1}{0 + \cos^{3} (0)}$

Step 7.2.7

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

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

Step 7.2.8

One to any power is one.

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

Step 7.2.9

Add $0$ and $1$ .

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

Step 7.3

Cancel the common factor of $1$ .

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Step 7.3.1

Cancel the common factor.

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

Step 7.3.2

Rewrite the expression.

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

Step 7.4

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

$\frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$