Calculus Examples

$lim_{x \to - 8} \frac{\sqrt{4 x^{6}}}{x^{3}}$

Step 1

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

$\frac{lim_{x \to - 8} \sqrt{4 x^{6}}}{lim_{x \to - 8} x^{3}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 8} 4 x^{6}}}{lim_{x \to - 8} x^{3}}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{4 lim_{x \to - 8} x^{6}}}{lim_{x \to - 8} x^{3}}$

Step 4

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

$\frac{\sqrt{4 {(lim_{x \to - 8} x)}^{6}}}{lim_{x \to - 8} x^{3}}$

Step 5

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

$\frac{\sqrt{4 {(lim_{x \to - 8} x)}^{6}}}{{(lim_{x \to - 8} x)}^{3}}$

Step 6

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

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Step 6.1

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

$\frac{\sqrt{4 {(- 8)}^{6}}}{{(lim_{x \to - 8} x)}^{3}}$

Step 6.2

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

$\frac{\sqrt{4 {(- 8)}^{6}}}{{(- 8)}^{3}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{\sqrt{4 \cdot 262144}}{{(- 8)}^{3}}$

Step 7.1.2

Multiply $4$ by $262144$ .

$\frac{\sqrt{1048576}}{{(- 8)}^{3}}$

Step 7.1.3

Rewrite $1048576$ as $1024^{2}$ .

$\frac{\sqrt{1024^{2}}}{{(- 8)}^{3}}$

Step 7.1.4

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

$\frac{1024}{{(- 8)}^{3}}$

Step 7.2

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

$\frac{1024}{- 512}$

Step 7.3

Divide $1024$ by $- 512$ .

$- 2$