Calculus Examples

$\frac{lim_{x \to - 8} \sqrt{x^{6} - 1}}{x^{5} - 1}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 8} x^{6} - 1}}{x^{5} - 1}$

Step 1.2

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

$\frac{\sqrt{lim_{x \to - 8} x^{6} - lim_{x \to - 8} 1}}{x^{5} - 1}$

Step 1.3

Move the exponent $6$ from $x^{6}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{{(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} 1}}{x^{5} - 1}$

Step 1.4

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

$\frac{\sqrt{{(lim_{x \to - 8} x)}^{6} - 1 \cdot 1}}{x^{5} - 1}$

Step 2

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

$\frac{\sqrt{{(- 8)}^{6} - 1 \cdot 1}}{x^{5} - 1}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $- 8$ to the power of $6$ .

$\frac{\sqrt{262144 - 1 \cdot 1}}{x^{5} - 1}$

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{\sqrt{262144 - 1}}{x^{5} - 1}$

Step 3.3

Subtract $1$ from $262144$ .

$\frac{\sqrt{262143}}{x^{5} - 1}$

Step 3.4

Rewrite $262143$ as $3^{2} \cdot 29127$ .

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Step 3.4.1

Factor $9$ out of $262143$ .

$\frac{\sqrt{9 (29127)}}{x^{5} - 1}$

Step 3.4.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 29127}}{x^{5} - 1}$

Step 3.5

Pull terms out from under the radical.

$\frac{3 \sqrt{29127}}{x^{5} - 1}$