Calculus Examples

$\frac{lim_{x \to 2} \sqrt{2 x^{3} + 4}}{\sqrt{x^{2} + 1}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2} 2 x^{3} + 4}}{\sqrt{x^{2} + 1}}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt{lim_{x \to 2} 2 x^{3} + lim_{x \to 2} 4}}{\sqrt{x^{2} + 1}}$

Step 1.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{2 lim_{x \to 2} x^{3} + lim_{x \to 2} 4}}{\sqrt{x^{2} + 1}}$

Step 1.4

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{2 {(lim_{x \to 2} x)}^{3} + lim_{x \to 2} 4}}{\sqrt{x^{2} + 1}}$

Step 1.5

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{2 {(lim_{x \to 2} x)}^{3} + 4}}{\sqrt{x^{2} + 1}}$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 \cdot 2^{3} + 4}}{\sqrt{x^{2} + 1}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $2$ by $2^{3}$ by adding the exponents.

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Step 3.1.1.1

Multiply $2$ by $2^{3}$ .

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Step 3.1.1.1.1

Raise $2$ to the power of $1$ .

$\frac{\sqrt{2^{1} \cdot 2^{3} + 4}}{\sqrt{x^{2} + 1}}$

Step 3.1.1.1.2

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

$\frac{\sqrt{2^{1 + 3} + 4}}{\sqrt{x^{2} + 1}}$

Step 3.1.1.2

Add $1$ and $3$ .

$\frac{\sqrt{2^{4} + 4}}{\sqrt{x^{2} + 1}}$

Step 3.1.2

Raise $2$ to the power of $4$ .

$\frac{\sqrt{16 + 4}}{\sqrt{x^{2} + 1}}$

Step 3.1.3

Add $16$ and $4$ .

$\frac{\sqrt{20}}{\sqrt{x^{2} + 1}}$

Step 3.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.1.4.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{\sqrt{x^{2} + 1}}$

Step 3.1.4.2

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

$\frac{\sqrt{2^{2} \cdot 5}}{\sqrt{x^{2} + 1}}$

Step 3.1.5

Pull terms out from under the radical.

$\frac{2 \sqrt{5}}{\sqrt{x^{2} + 1}}$

Step 3.2

Multiply $\frac{2 \sqrt{5}}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$\frac{2 \sqrt{5}}{\sqrt{x^{2} + 1}} \cdot \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{2 \sqrt{5}}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1} \sqrt{x^{2} + 1}}$

Step 3.3.2

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

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{1} \sqrt{x^{2} + 1}}$

Step 3.3.3

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

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{1} {\sqrt{x^{2} + 1}}^{1}}$

Step 3.3.4

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

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{1 + 1}}$

Step 3.3.5

Add $1$ and $1$ .

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{2}}$

Step 3.3.6

Rewrite ${\sqrt{x^{2} + 1}}^{2}$ as $x^{2} + 1$ .

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Step 3.3.6.1

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

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{({(x^{2} + 1)}^{\frac{1}{2}})}^{2}}$

Step 3.3.6.2

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

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 3.3.6.3

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

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{2}{2}}}$

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{2}{2}}}$

Step 3.3.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{1}}$

Step 3.3.6.5

Simplify.

$\frac{2 \sqrt{5} \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 3.4

Combine using the product rule for radicals.

$\frac{2 \sqrt{(x^{2} + 1) \cdot 5}}{x^{2} + 1}$

Step 3.5

Move $5$ to the left of $x^{2} + 1$ .

$\frac{2 \sqrt{5 (x^{2} + 1)}}{x^{2} + 1}$