Calculus Examples

$lim_{x \to 5} \frac{\sqrt{x + 11}}{\sqrt{x - 1}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} \sqrt{x + 11}}{lim_{x \to 5} \sqrt{x - 1}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 5} x + 11}}{lim_{x \to 5} \sqrt{x - 1}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + lim_{x \to 5} 11}}{lim_{x \to 5} \sqrt{x - 1}}$

Step 4

Evaluate the limit of $11$ which is constant as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + 11}}{lim_{x \to 5} \sqrt{x - 1}}$

Step 5

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 5} x + 11}}{\sqrt{lim_{x \to 5} x - 1}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + 11}}{\sqrt{lim_{x \to 5} x - lim_{x \to 5} 1}}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + 11}}{\sqrt{lim_{x \to 5} x - 1 \cdot 1}}$

Step 8

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sqrt{5 + 11}}{\sqrt{lim_{x \to 5} x - 1 \cdot 1}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sqrt{5 + 11}}{\sqrt{5 - 1 \cdot 1}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Add $5$ and $11$ .

$\frac{\sqrt{16}}{\sqrt{5 - 1 \cdot 1}}$

Step 9.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}}}{\sqrt{5 - 1 \cdot 1}}$

Step 9.1.3

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

$\frac{4}{\sqrt{5 - 1 \cdot 1}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 1$ by $1$ .

$\frac{4}{\sqrt{5 - 1}}$

Step 9.2.2

Subtract $1$ from $5$ .

$\frac{4}{\sqrt{4}}$

Step 9.2.3

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

$\frac{4}{\sqrt{2^{2}}}$

Step 9.2.4

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

$\frac{4}{2}$

Step 9.3

Divide $4$ by $2$ .

$2$