Calculus Examples

$\frac{lim_{x \to 0} \cos^{2} (4 x)}{x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $2$ from $\cos^{2} (4 x)$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} \cos (4 x))}^{2}}{x^{2}}$

Step 1.2

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

$\frac{\cos^{2} (lim_{x \to 0} 4 x)}{x^{2}}$

Step 1.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\cos^{2} (4 lim_{x \to 0} x)}{x^{2}}$

Step 2

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

$\frac{\cos^{2} (4 \cdot 0)}{x^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $4$ by $0$ .

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

Step 3.2

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

$\frac{1^{2}}{x^{2}}$

Step 3.3

One to any power is one.

$\frac{1}{x^{2}}$