Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Apply pythagorean identity.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

$\frac{0^{2}}{x}$

Step 3.2.2

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

$\frac{0}{x}$

Step 3.3

Divide $0$ by $x$ .

$0$