Calculus Examples

$lim_{x \to - 8} \frac{\sqrt{9 x^{10} - 1}}{- 4 x^{5}}$

Step 1

Move the term $\frac{1}{- 4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- 4} lim_{x \to - 8} \frac{\sqrt{9 x^{10} - 1}}{x^{5}}$

Step 2

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

$\frac{1}{- 4} \cdot \frac{lim_{x \to - 8} \sqrt{9 x^{10} - 1}}{lim_{x \to - 8} x^{5}}$

Step 3

Move the limit under the radical sign.

$\frac{1}{- 4} \cdot \frac{\sqrt{lim_{x \to - 8} 9 x^{10} - 1}}{lim_{x \to - 8} x^{5}}$

Step 4

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

$\frac{1}{- 4} \cdot \frac{\sqrt{lim_{x \to - 8} 9 x^{10} - lim_{x \to - 8} 1}}{lim_{x \to - 8} x^{5}}$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- 4} \cdot \frac{\sqrt{9 lim_{x \to - 8} x^{10} - lim_{x \to - 8} 1}}{lim_{x \to - 8} x^{5}}$

Step 6

Move the exponent $10$ from $x^{10}$ outside the limit using the Limits Power Rule.

$\frac{1}{- 4} \cdot \frac{\sqrt{9 {(lim_{x \to - 8} x)}^{10} - lim_{x \to - 8} 1}}{lim_{x \to - 8} x^{5}}$

Step 7

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

$\frac{1}{- 4} \cdot \frac{\sqrt{9 {(lim_{x \to - 8} x)}^{10} - 1 \cdot 1}}{lim_{x \to - 8} x^{5}}$

Step 8

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{1}{- 4} \cdot \frac{\sqrt{9 {(lim_{x \to - 8} x)}^{10} - 1 \cdot 1}}{{(lim_{x \to - 8} x)}^{5}}$

Step 9

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 9.1

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

$\frac{1}{- 4} \cdot \frac{\sqrt{9 {(- 8)}^{10} - 1 \cdot 1}}{{(lim_{x \to - 8} x)}^{5}}$

Step 9.2

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

$\frac{1}{- 4} \cdot \frac{\sqrt{9 {(- 8)}^{10} - 1 \cdot 1}}{{(- 8)}^{5}}$

Step 10

Simplify the answer.

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Step 10.1

Combine.

$\frac{1 \sqrt{9 {(- 8)}^{10} - 1 \cdot 1}}{- 4 {(- 8)}^{5}}$

Step 10.2

Multiply $\sqrt{9 {(- 8)}^{10} - 1 \cdot 1}$ by $1$ .

$\frac{\sqrt{9 {(- 8)}^{10} - 1 \cdot 1}}{- 4 {(- 8)}^{5}}$

Step 10.3

Raise $- 8$ to the power of $5$ .

$\frac{\sqrt{9 {(- 8)}^{10} - 1 \cdot 1}}{- 4 \cdot - 32768}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Raise $- 8$ to the power of $10$ .

$\frac{\sqrt{9 \cdot 1073741824 - 1 \cdot 1}}{- 4 \cdot - 32768}$

Step 10.4.2

Multiply $9$ by $1073741824$ .

$\frac{\sqrt{9663676416 - 1 \cdot 1}}{- 4 \cdot - 32768}$

Step 10.4.3

Multiply $- 1$ by $1$ .

$\frac{\sqrt{9663676416 - 1}}{- 4 \cdot - 32768}$

Step 10.4.4

Subtract $1$ from $9663676416$ .

$\frac{\sqrt{9663676415}}{- 4 \cdot - 32768}$

Step 10.5

Multiply $- 4$ by $- 32768$ .

$\frac{\sqrt{9663676415}}{131072}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{9663676415}}{131072}$

Decimal Form:

$0.74999999 \dots$