Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

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

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

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

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

Step 1.5

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

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

Step 1.6

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

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

Step 1.7

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

$\sqrt{1 + \frac{1}{64}}$

Step 3.2

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{64}{64} + \frac{1}{64}}$

Step 3.3

Combine the numerators over the common denominator.

$\sqrt{\frac{64 + 1}{64}}$

Step 3.4

Add $64$ and $1$ .

$\sqrt{\frac{65}{64}}$

Step 3.5

Rewrite $\sqrt{\frac{65}{64}}$ as $\frac{\sqrt{65}}{\sqrt{64}}$ .

$\frac{\sqrt{65}}{\sqrt{64}}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{65}}{\sqrt{8^{2}}}$

Step 3.6.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{65}}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{65}}{8}$

Decimal Form:

$1.00778221 \dots$