Calculus Examples

$\frac{lim_{x \to 0} 9 - 9 \cos (x)}{x^{2}}$

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 $0$ .

$\frac{lim_{x \to 0} 9 - lim_{x \to 0} 9 \cos (x)}{x^{2}}$

Step 1.2

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$\frac{9 - lim_{x \to 0} 9 \cos (x)}{x^{2}}$

Step 1.3

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 - 9 lim_{x \to 0} \cos (x)}{x^{2}}$

Step 1.4

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

$\frac{9 - 9 \cos (lim_{x \to 0} x)}{x^{2}}$

Step 2

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

$\frac{9 - 9 \cos (0)}{x^{2}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{9 - 9 \cdot 1}{x^{2}}$

Step 3.1.2

Multiply $- 9$ by $1$ .

$\frac{9 - 9}{x^{2}}$

Step 3.1.3

Subtract $9$ from $9$ .

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

Step 3.2

Divide $0$ by $x^{2}$ .

$0$