Calculus Examples

$lim_{x \to 0} \frac{16 \cos (16 x)}{26 \cos (26 x)}$

Step 1

Cancel the common factor of $16$ and $26$ .

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Step 1.1

Factor $2$ out of $16 \cos (16 x)$ .

$lim_{x \to 0} \frac{2 (8 \cos (16 x))}{26 \cos (26 x)}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $26 \cos (26 x)$ .

$lim_{x \to 0} \frac{2 (8 \cos (16 x))}{2 (13 \cos (26 x))}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{2 (8 \cos (16 x))}{2 (13 \cos (26 x))}$

Step 1.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{8 \cos (16 x)}{13 \cos (26 x)}$

Step 2

Move the term $\frac{8}{13}$ outside of the limit because it is constant with respect to $x$ .

$\frac{8}{13} lim_{x \to 0} \frac{\cos (16 x)}{\cos (26 x)}$

Step 3

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

$\frac{8}{13} \cdot \frac{lim_{x \to 0} \cos (16 x)}{lim_{x \to 0} \cos (26 x)}$

Step 4

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

$\frac{8}{13} \cdot \frac{\cos (lim_{x \to 0} 16 x)}{lim_{x \to 0} \cos (26 x)}$

Step 5

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\frac{8}{13} \cdot \frac{\cos (16 lim_{x \to 0} x)}{lim_{x \to 0} \cos (26 x)}$

Step 6

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

$\frac{8}{13} \cdot \frac{\cos (16 lim_{x \to 0} x)}{\cos (lim_{x \to 0} 26 x)}$

Step 7

Move the term $26$ outside of the limit because it is constant with respect to $x$ .

$\frac{8}{13} \cdot \frac{\cos (16 lim_{x \to 0} x)}{\cos (26 lim_{x \to 0} x)}$

Step 8

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

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Step 8.1

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

$\frac{8}{13} \cdot \frac{\cos (16 \cdot 0)}{\cos (26 lim_{x \to 0} x)}$

Step 8.2

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

$\frac{8}{13} \cdot \frac{\cos (16 \cdot 0)}{\cos (26 \cdot 0)}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $16$ by $0$ .

$\frac{8}{13} \cdot \frac{\cos (0)}{\cos (26 \cdot 0)}$

Step 9.1.2

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

$\frac{8}{13} \cdot \frac{1}{\cos (26 \cdot 0)}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $26$ by $0$ .

$\frac{8}{13} \cdot \frac{1}{\cos (0)}$

Step 9.2.2

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

$\frac{8}{13} \cdot \frac{1}{1}$

Step 9.3

Cancel the common factor of $1$ .

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Step 9.3.1

Cancel the common factor.

$\frac{8}{13} \cdot \frac{1}{1}$

Step 9.3.2

Rewrite the expression.

$\frac{8}{13} \cdot 1$

Step 9.4

Multiply $\frac{8}{13}$ by $1$ .

$\frac{8}{13}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{13}$

Decimal Form:

$0.\overline{615384}$