Precalculus Examples

$\frac{lim_{x \to 0} (\sqrt{x^{2} + 6 x + 13}) - \sqrt{13}}{x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sqrt{x^{2} + 6 x + 13} - lim_{x \to 0} \sqrt{13}}{x}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} x^{2} + 6 x + 13} - lim_{x \to 0} \sqrt{13}}{x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x^{2} + lim_{x \to 0} 6 x + lim_{x \to 0} 13} - lim_{x \to 0} \sqrt{13}}{x}$

Step 4

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

$\frac{\sqrt{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 6 x + lim_{x \to 0} 13} - lim_{x \to 0} \sqrt{13}}{x}$

Step 5

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{{(lim_{x \to 0} x)}^{2} + 6 lim_{x \to 0} x + lim_{x \to 0} 13} - lim_{x \to 0} \sqrt{13}}{x}$

Step 6

Evaluate the limit of $13$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{{(lim_{x \to 0} x)}^{2} + 6 lim_{x \to 0} x + 13} - lim_{x \to 0} \sqrt{13}}{x}$

Step 7

Evaluate the limit of $\sqrt{13}$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{{(lim_{x \to 0} x)}^{2} + 6 lim_{x \to 0} x + 13} - \sqrt{13}}{x}$

Step 8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt{0^{2} + 6 lim_{x \to 0} x + 13} - \sqrt{13}}{x}$

Step 8.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt{0^{2} + 6 \cdot 0 + 13} - \sqrt{13}}{x}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{\sqrt{0 + 6 \cdot 0 + 13} - \sqrt{13}}{x}$

Step 9.1.2

Multiply $6$ by $0$ .

$\frac{\sqrt{0 + 0 + 13} - \sqrt{13}}{x}$

Step 9.1.3

Add $0$ and $0$ .

$\frac{\sqrt{0 + 13} - \sqrt{13}}{x}$

Step 9.1.4

Add $0$ and $13$ .

$\frac{\sqrt{13} - \sqrt{13}}{x}$

Step 9.1.5

Subtract $\sqrt{13}$ from $\sqrt{13}$ .

$\frac{0}{x}$

Step 9.2

Divide $0$ by $x$ .

$0$