Calculus Examples

$lim_{x \to - 8} \frac{\sqrt{9 x^{6} - x}}{x^{3} + x}$

Step 1

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

$\frac{lim_{x \to - 8} \sqrt{9 x^{6} - x}}{lim_{x \to - 8} x^{3} + x}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 8} 9 x^{6} - x}}{lim_{x \to - 8} x^{3} + x}$

Step 3

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

$\frac{\sqrt{lim_{x \to - 8} 9 x^{6} - lim_{x \to - 8} x}}{lim_{x \to - 8} x^{3} + x}$

Step 4

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

$\frac{\sqrt{9 lim_{x \to - 8} x^{6} - lim_{x \to - 8} x}}{lim_{x \to - 8} x^{3} + x}$

Step 5

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

$\frac{\sqrt{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}}{lim_{x \to - 8} x^{3} + x}$

Step 6

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

$\frac{\sqrt{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}}{lim_{x \to - 8} x^{3} + lim_{x \to - 8} x}$

Step 7

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{9 {(lim_{x \to - 8} x)}^{6} - lim_{x \to - 8} x}}{{(lim_{x \to - 8} x)}^{3} + lim_{x \to - 8} x}$

Step 8

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

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Step 8.1

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

$\frac{\sqrt{9 {(- 8)}^{6} - lim_{x \to - 8} x}}{{(lim_{x \to - 8} x)}^{3} + lim_{x \to - 8} x}$

Step 8.2

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

$\frac{\sqrt{9 {(- 8)}^{6} + 8}}{{(lim_{x \to - 8} x)}^{3} + lim_{x \to - 8} x}$

Step 8.3

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

$\frac{\sqrt{9 {(- 8)}^{6} + 8}}{{(- 8)}^{3} + lim_{x \to - 8} x}$

Step 8.4

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

$\frac{\sqrt{9 {(- 8)}^{6} + 8}}{{(- 8)}^{3} - 8}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{\sqrt{9 \cdot 262144 + 8}}{{(- 8)}^{3} - 8}$

Step 9.1.2

Multiply $9$ by $262144$ .

$\frac{\sqrt{2359296 + 8}}{{(- 8)}^{3} - 8}$

Step 9.1.3

Add $2359296$ and $8$ .

$\frac{\sqrt{2359304}}{{(- 8)}^{3} - 8}$

Step 9.1.4

Rewrite $2359304$ as $2^{2} \cdot 589826$ .

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Step 9.1.4.1

Factor $4$ out of $2359304$ .

$\frac{\sqrt{4 (589826)}}{{(- 8)}^{3} - 8}$

Step 9.1.4.2

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

$\frac{\sqrt{2^{2} \cdot 589826}}{{(- 8)}^{3} - 8}$

Step 9.1.5

Pull terms out from under the radical.

$\frac{2 \sqrt{589826}}{{(- 8)}^{3} - 8}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Raise $- 8$ to the power of $3$ .

$\frac{2 \sqrt{589826}}{- 512 - 8}$

Step 9.2.2

Subtract $8$ from $- 512$ .

$\frac{2 \sqrt{589826}}{- 520}$

Step 9.3

Cancel the common factor of $2$ and $- 520$ .

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Step 9.3.1

Factor $2$ out of $2 \sqrt{589826}$ .

$\frac{2 (\sqrt{589826})}{- 520}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $- 520$ .

$\frac{2 \sqrt{589826}}{2 \cdot - 260}$

Step 9.3.2.2

Cancel the common factor.

$\frac{2 \sqrt{589826}}{2 \cdot - 260}$

Step 9.3.2.3

Rewrite the expression.

$\frac{\sqrt{589826}}{- 260}$

Step 9.4

Move the negative in front of the fraction.

$- \frac{\sqrt{589826}}{260}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{589826}}{260}$

Decimal Form:

$- 2.95385116 \dots$