Calculus Examples

$\frac{lim_{x \to 8} \sqrt{x^{2} + 7}}{\sqrt{7 x^{2} + 1}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} x^{2} + 7}}{\sqrt{7 x^{2} + 1}}$

Step 1.2

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

$\frac{\sqrt{lim_{x \to 8} x^{2} + lim_{x \to 8} 7}}{\sqrt{7 x^{2} + 1}}$

Step 1.3

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

$\frac{\sqrt{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 7}}{\sqrt{7 x^{2} + 1}}$

Step 1.4

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{{(lim_{x \to 8} x)}^{2} + 7}}{\sqrt{7 x^{2} + 1}}$

Step 2

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

$\frac{\sqrt{8^{2} + 7}}{\sqrt{7 x^{2} + 1}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $8$ to the power of $2$ .

$\frac{\sqrt{64 + 7}}{\sqrt{7 x^{2} + 1}}$

Step 3.1.2

Add $64$ and $7$ .

$\frac{\sqrt{71}}{\sqrt{7 x^{2} + 1}}$

Step 3.2

Multiply $\frac{\sqrt{71}}{\sqrt{7 x^{2} + 1}}$ by $\frac{\sqrt{7 x^{2} + 1}}{\sqrt{7 x^{2} + 1}}$ .

$\frac{\sqrt{71}}{\sqrt{7 x^{2} + 1}} \cdot \frac{\sqrt{7 x^{2} + 1}}{\sqrt{7 x^{2} + 1}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{\sqrt{71}}{\sqrt{7 x^{2} + 1}}$ by $\frac{\sqrt{7 x^{2} + 1}}{\sqrt{7 x^{2} + 1}}$ .

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{\sqrt{7 x^{2} + 1} \sqrt{7 x^{2} + 1}}$

Step 3.3.2

Raise $\sqrt{7 x^{2} + 1}$ to the power of $1$ .

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{\sqrt{7 x^{2} + 1}}^{1} \sqrt{7 x^{2} + 1}}$

Step 3.3.3

Raise $\sqrt{7 x^{2} + 1}$ to the power of $1$ .

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{\sqrt{7 x^{2} + 1}}^{1} {\sqrt{7 x^{2} + 1}}^{1}}$

Step 3.3.4

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

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{\sqrt{7 x^{2} + 1}}^{1 + 1}}$

Step 3.3.5

Add $1$ and $1$ .

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{\sqrt{7 x^{2} + 1}}^{2}}$

Step 3.3.6

Rewrite ${\sqrt{7 x^{2} + 1}}^{2}$ as $7 x^{2} + 1$ .

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Step 3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x^{2} + 1}$ as ${(7 x^{2} + 1)}^{\frac{1}{2}}$ .

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{({(7 x^{2} + 1)}^{\frac{1}{2}})}^{2}}$

Step 3.3.6.2

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

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{(7 x^{2} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 3.3.6.3

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

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{(7 x^{2} + 1)}^{\frac{2}{2}}}$

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.

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{(7 x^{2} + 1)}^{\frac{2}{2}}}$

Step 3.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{{(7 x^{2} + 1)}^{1}}$

Step 3.3.6.5

Simplify.

$\frac{\sqrt{71} \sqrt{7 x^{2} + 1}}{7 x^{2} + 1}$

Step 3.4

Combine using the product rule for radicals.

$\frac{\sqrt{71 (7 x^{2} + 1)}}{7 x^{2} + 1}$