Calculus Examples

$lim_{x \to 0} \frac{1 - \cos (x) + \tan (5 x)}{1 - \cos (x) - \tan (x)}$

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.

$\frac{lim_{x \to 0} 1 - \cos (x) + \tan (5 x)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

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 $0$ .

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \tan (5 x)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.2

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

$\frac{1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \tan (5 x)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.3

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

$\frac{1 - \cos (lim_{x \to 0} x) + lim_{x \to 0} \tan (5 x)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.4

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

$\frac{1 - \cos (lim_{x \to 0} x) + \tan (lim_{x \to 0} 5 x)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 - \cos (lim_{x \to 0} x) + \tan (5 lim_{x \to 0} x)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$\frac{1 - \cos (0) + \tan (5 lim_{x \to 0} x)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.6.2

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

$\frac{1 - \cos (0) + \tan (5 \cdot 0)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

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

$\frac{1 - 1 \cdot 1 + \tan (5 \cdot 0)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.7.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + \tan (5 \cdot 0)}{lim_{x \to 0} 1 - \cos (x) - \tan (x)}$

Step 1.1.2.7.1.3

Multiply $5$ by $0$ .

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

Step 1.1.2.7.1.4

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

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

Step 1.1.2.7.2

Subtract $1$ from $1$ .

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

Step 1.1.2.7.3

Add $0$ and $0$ .

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

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 $0$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

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

Step 1.1.3.5.2

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

$\frac{0}{1 - \cos (0) - \tan (0)}$

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

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

$\frac{0}{1 - 1 \cdot 1 - \tan (0)}$

Step 1.1.3.6.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.6.1.3

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

$\frac{0}{1 - 1 - 0}$

Step 1.1.3.6.1.4

Multiply $- 1$ by $0$ .

$\frac{0}{1 - 1 + 0}$

Step 1.1.3.6.2

Subtract $1$ from $1$ .

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

Step 1.1.3.6.3

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.6.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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.

$lim_{x \to 0} \frac{1 - \cos (x) + \tan (5 x)}{1 - \cos (x) - \tan (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (x) + \tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (x) + \tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.2

By the Sum Rule, the derivative of $1 - \cos (x) + \tan (5 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [\tan (5 x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [\tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.3

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

$lim_{x \to 0} \frac{0 + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [\tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 0} \frac{0 - \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [\tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{0 - - \sin (x) + \frac{d}{d x} [\tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.4.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{0 + 1 \sin (x) + \frac{d}{d x} [\tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.4.4

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

$lim_{x \to 0} \frac{0 + \sin (x) + \frac{d}{d x} [\tan (5 x)]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.5

Evaluate $\frac{d}{d x} [\tan (5 x)]$ .

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Step 1.3.5.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 5 x$ .

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Step 1.3.5.1.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to 0} \frac{0 + \sin (x) + \frac{d}{d u} [\tan (u)] \frac{d}{d x} [5 x]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.5.1.2

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

$lim_{x \to 0} \frac{0 + \sin (x) + \sec^{2} (u) \frac{d}{d x} [5 x]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.5.1.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to 0} \frac{0 + \sin (x) + \sec^{2} (5 x) \frac{d}{d x} [5 x]}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.5.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{0 + \sin (x) + \sec^{2} (5 x) (5 \frac{d}{d x} [x])}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.5.3

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

$lim_{x \to 0} \frac{0 + \sin (x) + \sec^{2} (5 x) (5 \cdot 1)}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.5.4

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{0 + \sin (x) + \sec^{2} (5 x) \cdot 5}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.5.5

Move $5$ to the left of $\sec^{2} (5 x)$ .

$lim_{x \to 0} \frac{0 + \sin (x) + 5 \sec^{2} (5 x)}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.6

Simplify.

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Step 1.3.6.1

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

$lim_{x \to 0} \frac{\sin (x) + 5 \sec^{2} (5 x)}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.6.2

Simplify each term.

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Step 1.3.6.2.1

Rewrite $\sec (5 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\sin (x) + 5 {(\frac{1}{\cos (5 x)})}^{2}}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.6.2.2

Apply the product rule to $\frac{1}{\cos (5 x)}$ .

$lim_{x \to 0} \frac{\sin (x) + 5 \frac{1^{2}}{\cos^{2} (5 x)}}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.6.2.3

One to any power is one.

$lim_{x \to 0} \frac{\sin (x) + 5 \frac{1}{\cos^{2} (5 x)}}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.6.2.4

Combine $5$ and $\frac{1}{\cos^{2} (5 x)}$ .

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{\frac{d}{d x} [1 - \cos (x) - \tan (x)]}$

Step 1.3.7

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.8

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{0 + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.9

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

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Step 1.3.9.1

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{0 - \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.9.2

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{0 - - \sin (x) + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.9.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{0 + 1 \sin (x) + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.9.4

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{0 + \sin (x) + \frac{d}{d x} [- \tan (x)]}$

Step 1.3.10

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

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Step 1.3.10.1

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{0 + \sin (x) - \frac{d}{d x} [\tan (x)]}$

Step 1.3.10.2

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{0 + \sin (x) - \sec^{2} (x)}$

Step 1.3.11

Simplify.

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Step 1.3.11.1

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{\sin (x) - \sec^{2} (x)}$

Step 1.3.11.2

Simplify each term.

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Step 1.3.11.2.1

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{\sin (x) - {(\frac{1}{\cos (x)})}^{2}}$

Step 1.3.11.2.2

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

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{\sin (x) - \frac{1^{2}}{\cos^{2} (x)}}$

Step 1.3.11.2.3

One to any power is one.

$lim_{x \to 0} \frac{\sin (x) + \frac{5}{\cos^{2} (5 x)}}{\sin (x) - \frac{1}{\cos^{2} (x)}}$

Step 1.4

Combine terms.

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Step 1.4.1

To write $\sin (x)$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (5 x)}{\cos^{2} (5 x)}$ .

$lim_{x \to 0} \frac{\frac{\sin (x) \cos^{2} (5 x)}{\cos^{2} (5 x)} + \frac{5}{\cos^{2} (5 x)}}{\sin (x) - \frac{1}{\cos^{2} (x)}}$

Step 1.4.2

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{\sin (x) \cos^{2} (5 x) + 5}{\cos^{2} (5 x)}}{\sin (x) - \frac{1}{\cos^{2} (x)}}$

Step 1.4.3

To write $\sin (x)$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

$lim_{x \to 0} \frac{\frac{\sin (x) \cos^{2} (5 x) + 5}{\cos^{2} (5 x)}}{\frac{\sin (x) \cos^{2} (x)}{\cos^{2} (x)} - \frac{1}{\cos^{2} (x)}}$

Step 1.4.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{\sin (x) \cos^{2} (5 x) + 5}{\cos^{2} (5 x)}}{\frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{lim_{x \to 0} \frac{\sin (x) \cos^{2} (5 x) + 5}{\cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.2

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

$\frac{\frac{lim_{x \to 0} \sin (x) \cos^{2} (5 x) + 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.3

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

$\frac{\frac{lim_{x \to 0} \sin (x) \cos^{2} (5 x) + lim_{x \to 0} 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.4

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

$\frac{\frac{lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \cos^{2} (5 x) + lim_{x \to 0} 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.5

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos^{2} (5 x) + lim_{x \to 0} 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.6

Move the exponent $2$ from $\cos^{2} (5 x)$ outside the limit using the Limits Power Rule.

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot {(lim_{x \to 0} \cos (5 x))}^{2} + lim_{x \to 0} 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.7

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (lim_{x \to 0} 5 x) + lim_{x \to 0} 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.8

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + lim_{x \to 0} 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.9

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{lim_{x \to 0} \cos^{2} (5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.10

Move the exponent $2$ from $\cos^{2} (5 x)$ outside the limit using the Limits Power Rule.

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{{(lim_{x \to 0} \cos (5 x))}^{2}}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.11

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (lim_{x \to 0} 5 x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.12

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{lim_{x \to 0} \frac{\sin (x) \cos^{2} (x) - 1}{\cos^{2} (x)}}$

Step 2.13

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{lim_{x \to 0} \sin (x) \cos^{2} (x) - 1}{lim_{x \to 0} \cos^{2} (x)}}$

Step 2.14

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{lim_{x \to 0} \sin (x) \cos^{2} (x) - lim_{x \to 0} 1}{lim_{x \to 0} \cos^{2} (x)}}$

Step 2.15

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \cos^{2} (x) - lim_{x \to 0} 1}{lim_{x \to 0} \cos^{2} (x)}}$

Step 2.16

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{\sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos^{2} (x) - lim_{x \to 0} 1}{lim_{x \to 0} \cos^{2} (x)}}$

Step 2.17

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{\sin (lim_{x \to 0} x) \cdot {(lim_{x \to 0} \cos (x))}^{2} - lim_{x \to 0} 1}{lim_{x \to 0} \cos^{2} (x)}}$

Step 2.18

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (lim_{x \to 0} x) - lim_{x \to 0} 1}{lim_{x \to 0} \cos^{2} (x)}}$

Step 2.19

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

$\frac{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (lim_{x \to 0} x) - 1 \cdot 1}{lim_{x \to 0} \cos^{2} (x)}}$

Step 2.20

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

Step 2.21

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

Step 3

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

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Step 3.1

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

$\frac{\frac{\sin (0) \cdot \cos^{2} (5 lim_{x \to 0} x) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (lim_{x \to 0} x) - 1 \cdot 1}{\cos^{2} (lim_{x \to 0} x)}}$

Step 3.2

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

$\frac{\frac{\sin (0) \cdot \cos^{2} (5 \cdot 0) + 5}{\cos^{2} (5 lim_{x \to 0} x)}}{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (lim_{x \to 0} x) - 1 \cdot 1}{\cos^{2} (lim_{x \to 0} x)}}$

Step 3.3

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

$\frac{\frac{\sin (0) \cdot \cos^{2} (5 \cdot 0) + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (lim_{x \to 0} x) \cdot \cos^{2} (lim_{x \to 0} x) - 1 \cdot 1}{\cos^{2} (lim_{x \to 0} x)}}$

Step 3.4

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

$\frac{\frac{\sin (0) \cdot \cos^{2} (5 \cdot 0) + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (lim_{x \to 0} x) - 1 \cdot 1}{\cos^{2} (lim_{x \to 0} x)}}$

Step 3.5

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

$\frac{\frac{\sin (0) \cdot \cos^{2} (5 \cdot 0) + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (lim_{x \to 0} x)}}$

Step 3.6

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

$\frac{\frac{\sin (0) \cdot \cos^{2} (5 \cdot 0) + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$\frac{\frac{0 \cdot \cos^{2} (5 \cdot 0) + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.1.2

Multiply $5$ by $0$ .

$\frac{\frac{0 \cdot \cos^{2} (0) + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.1.3

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

$\frac{\frac{0 \cdot 1^{2} + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.1.4

One to any power is one.

$\frac{\frac{0 \cdot 1 + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.1.5

Multiply $0$ by $1$ .

$\frac{\frac{0 + 5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.1.6

Add $0$ and $5$ .

$\frac{\frac{5}{\cos^{2} (5 \cdot 0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $5$ by $0$ .

$\frac{\frac{5}{\cos^{2} (0)}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.2.2

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

$\frac{\frac{5}{1^{2}}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.2.3

One to any power is one.

$\frac{\frac{5}{1}}{\frac{\sin (0) \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

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

$\frac{\frac{5}{1}}{\frac{0 \cdot \cos^{2} (0) - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.3.2

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

$\frac{\frac{5}{1}}{\frac{0 \cdot 1^{2} - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.3.3

One to any power is one.

$\frac{\frac{5}{1}}{\frac{0 \cdot 1 - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.3.4

Multiply $0$ by $1$ .

$\frac{\frac{5}{1}}{\frac{0 - 1 \cdot 1}{\cos^{2} (0)}}$

Step 4.3.5

Multiply $- 1$ by $1$ .

$\frac{\frac{5}{1}}{\frac{0 - 1}{\cos^{2} (0)}}$

Step 4.3.6

Subtract $1$ from $0$ .

$\frac{\frac{5}{1}}{\frac{- 1}{\cos^{2} (0)}}$

Step 4.4

Simplify the denominator.

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Step 4.4.1

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

$\frac{\frac{5}{1}}{\frac{- 1}{1^{2}}}$

Step 4.4.2

One to any power is one.

$\frac{\frac{5}{1}}{\frac{- 1}{1}}$

Step 4.5

Divide $5$ by $1$ .

$\frac{5}{\frac{- 1}{1}}$

Step 4.6

Divide $- 1$ by $1$ .

$\frac{5}{- 1}$

Step 4.7

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$- 1 \cdot 5$

Step 4.8

Multiply $- 1$ by $5$ .

$- 5$