Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

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

Step 2

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

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

Step 3

Move the limit under the radical sign.

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

Step 4

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

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

Step 4.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\cos (\sqrt{8})}{8}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.1.1.1

Factor $4$ out of $8$ .

$\frac{\cos (\sqrt{4 (2)})}{8}$

Step 5.1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\cos (\sqrt{2^{2} \cdot 2})}{8}$

Step 5.1.2

Pull terms out from under the radical.

$\frac{\cos (2 \sqrt{2})}{8}$

Step 5.1.3

Evaluate $\cos (2 \sqrt{2})$ .

$\frac{- 0.95136312}{8}$

Step 5.2

Divide $- 0.95136312$ by $8$ .

$- 0.11892039$