Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

$\frac{\cos^{2} (0)}{\cos (x^{2})}$

Step 3

Simplify the answer.

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Step 3.1

Factor $\cos (0)$ out of $\cos^{2} (0)$ .

$\frac{\cos (0) \cos (0)}{\cos (x^{2})}$

Step 3.2

Separate fractions.

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

Step 3.3

Rewrite $\frac{\cos (0)}{\cos (x^{2})}$ as a product.

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

Step 3.4

Write $\cos (0)$ as a fraction with denominator $1$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Divide $\cos (0)$ by $1$ .

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

Step 3.5.2

Convert from $\frac{1}{\cos (x^{2})}$ to $\sec (x^{2})$ .

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

Step 3.6

Divide $\cos (0)$ by $1$ .

$\cos (0) (\cos (0) \sec (x^{2}))$

Step 3.7

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

$1 (\cos (0) \sec (x^{2}))$

Step 3.8

Multiply $\cos (0) \sec (x^{2})$ by $1$ .

$\cos (0) \sec (x^{2})$

Step 3.9

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

$1 \sec (x^{2})$

Step 3.10

Multiply $\sec (x^{2})$ by $1$ .

$\sec (x^{2})$