Calculus Examples

$\frac{lim_{x \to 5} \sqrt{x^{2} + 11} - 6}{x - 5}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 5} \sqrt{x^{2} + 11} - lim_{x \to 5} 6}{x - 5}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 5} x^{2} + 11} - lim_{x \to 5} 6}{x - 5}$

Step 1.3

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

$\frac{\sqrt{lim_{x \to 5} x^{2} + lim_{x \to 5} 11} - lim_{x \to 5} 6}{x - 5}$

Step 1.4

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

$\frac{\sqrt{{(lim_{x \to 5} x)}^{2} + lim_{x \to 5} 11} - lim_{x \to 5} 6}{x - 5}$

Step 1.5

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

$\frac{\sqrt{{(lim_{x \to 5} x)}^{2} + 11} - lim_{x \to 5} 6}{x - 5}$

Step 1.6

Evaluate the limit of $6$ which is constant as $x$ approaches $5$ .

$\frac{\sqrt{{(lim_{x \to 5} x)}^{2} + 11} - 1 \cdot 6}{x - 5}$

Step 2

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

$\frac{\sqrt{5^{2} + 11} - 1 \cdot 6}{x - 5}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $5$ to the power of $2$ .

$\frac{\sqrt{25 + 11} - 1 \cdot 6}{x - 5}$

Step 3.1.2

Add $25$ and $11$ .

$\frac{\sqrt{36} - 1 \cdot 6}{x - 5}$

Step 3.1.3

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2}} - 1 \cdot 6}{x - 5}$

Step 3.1.4

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

$\frac{6 - 1 \cdot 6}{x - 5}$

Step 3.1.5

Multiply $- 1$ by $6$ .

$\frac{6 - 6}{x - 5}$

Step 3.1.6

Subtract $6$ from $6$ .

$\frac{0}{x - 5}$

Step 3.2

Divide $0$ by $x - 5$ .

$0$