Calculus Examples

$lim_{x \to 8} \frac{\cos^{2} (x)}{\sqrt{x}}$

Step 1

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

$\frac{lim_{x \to 8} \cos^{2} (x)}{lim_{x \to 8} \sqrt{x}}$

Step 2

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

$\frac{{(lim_{x \to 8} \cos (x))}^{2}}{lim_{x \to 8} \sqrt{x}}$

Step 3

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

$\frac{\cos^{2} (lim_{x \to 8} x)}{lim_{x \to 8} \sqrt{x}}$

Step 4

Move the limit under the radical sign.

$\frac{\cos^{2} (lim_{x \to 8} x)}{\sqrt{lim_{x \to 8} x}}$

Step 5

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

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Step 5.1

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

$\frac{\cos^{2} (8)}{\sqrt{lim_{x \to 8} x}}$

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Evaluate $\cos (8)$ .

$\frac{{0.99026806}^{2}}{\sqrt{8}}$

Step 6.1.2

Raise $0.99026806$ to the power of $2$ .

$\frac{0.98063084}{\sqrt{8}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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Step 6.2.1.1

Factor $4$ out of $8$ .

$\frac{0.98063084}{\sqrt{4 (2)}}$

Step 6.2.1.2

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

$\frac{0.98063084}{\sqrt{2^{2} \cdot 2}}$

Step 6.2.2

Pull terms out from under the radical.

$\frac{0.98063084}{2 \sqrt{2}}$

Step 6.3

Multiply $\frac{0.98063084}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{0.98063084}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{0.98063084}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{0.98063084 \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 6.4.2

Move $\sqrt{2}$ .

$\frac{0.98063084 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 6.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{0.98063084 \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 6.4.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{0.98063084 \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 6.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{0.98063084 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 6.4.6

Add $1$ and $1$ .

$\frac{0.98063084 \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 6.4.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{0.98063084 \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 6.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{0.98063084 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 6.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{0.98063084 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 6.4.7.4

Cancel the common factor of $2$ .

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Step 6.4.7.4.1

Cancel the common factor.

$\frac{0.98063084 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 6.4.7.4.2

Rewrite the expression.

$\frac{0.98063084 \sqrt{2}}{2 \cdot 2^{1}}$

Step 6.4.7.5

Evaluate the exponent.

$\frac{0.98063084 \sqrt{2}}{2 \cdot 2}$

Step 6.5

Multiply $0.98063084$ by $\sqrt{2}$ .

$\frac{1.38682144}{2 \cdot 2}$

Step 6.6

Multiply $2$ by $2$ .

$\frac{1.38682144}{4}$

Step 6.7

Divide $1.38682144$ by $4$ .

$0.34670536$