Calculus Examples

$lim_{x \to 0} \frac{\cos (x)}{x + 8}$

Step 1

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

$\frac{lim_{x \to 0} \cos (x)}{lim_{x \to 0} x + 8}$

Step 2

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

$\frac{\cos (lim_{x \to 0} x)}{lim_{x \to 0} x + 8}$

Step 3

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

$\frac{\cos (lim_{x \to 0} x)}{lim_{x \to 0} x + lim_{x \to 0} 8}$

Step 4

Evaluate the limit of $8$ which is constant as $x$ approaches $0$ .

$\frac{\cos (lim_{x \to 0} x)}{lim_{x \to 0} x + 8}$

Step 5

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

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Step 5.1

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

$\frac{\cos (0)}{lim_{x \to 0} x + 8}$

Step 5.2

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

$\frac{\cos (0)}{0 + 8}$

Step 6

Simplify the answer.

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Step 6.1

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

$\frac{1}{0 + 8}$

Step 6.2

Add $0$ and $8$ .

$\frac{1}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{8}$

Decimal Form:

$0.125$