Calculus Examples

$lim_{x \to - 8} \frac{\sqrt[3]{x}}{p} + \frac{x^{3}}{\sqrt{x + 44}}$

Step 1

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

$lim_{x \to - 8} \frac{\sqrt[3]{x}}{p} + lim_{x \to - 8} \frac{x^{3}}{\sqrt{x + 44}}$

Step 2

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

$\frac{1}{p} lim_{x \to - 8} \sqrt[3]{x} + lim_{x \to - 8} \frac{x^{3}}{\sqrt{x + 44}}$

Step 3

Move the limit under the radical sign.

$\frac{1}{p} \sqrt[3]{lim_{x \to - 8} x} + lim_{x \to - 8} \frac{x^{3}}{\sqrt{x + 44}}$

Step 4

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

$\frac{1}{p} \sqrt[3]{lim_{x \to - 8} x} + \frac{lim_{x \to - 8} x^{3}}{lim_{x \to - 8} \sqrt{x + 44}}$

Step 5

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

$\frac{1}{p} \sqrt[3]{lim_{x \to - 8} x} + \frac{{(lim_{x \to - 8} x)}^{3}}{lim_{x \to - 8} \sqrt{x + 44}}$

Step 6

Move the limit under the radical sign.

$\frac{1}{p} \sqrt[3]{lim_{x \to - 8} x} + \frac{{(lim_{x \to - 8} x)}^{3}}{\sqrt{lim_{x \to - 8} x + 44}}$

Step 7

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

$\frac{1}{p} \sqrt[3]{lim_{x \to - 8} x} + \frac{{(lim_{x \to - 8} x)}^{3}}{\sqrt{lim_{x \to - 8} x + lim_{x \to - 8} 44}}$

Step 8

Evaluate the limit of $44$ which is constant as $x$ approaches $- 8$ .

$\frac{1}{p} \sqrt[3]{lim_{x \to - 8} x} + \frac{{(lim_{x \to - 8} x)}^{3}}{\sqrt{lim_{x \to - 8} x + 44}}$

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 $x$ .

$\frac{1}{p} \sqrt[3]{- 8} + \frac{{(lim_{x \to - 8} x)}^{3}}{\sqrt{lim_{x \to - 8} x + 44}}$

Step 9.2

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

$\frac{1}{p} \sqrt[3]{- 8} + \frac{{(- 8)}^{3}}{\sqrt{lim_{x \to - 8} x + 44}}$

Step 9.3

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

$\frac{1}{p} \sqrt[3]{- 8} + \frac{{(- 8)}^{3}}{\sqrt{- 8 + 44}}$

Step 10

Simplify each term.

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Step 10.1

Rewrite $- 8$ as ${(- 2)}^{3}$ .

$\frac{1}{p} \sqrt[3]{{(- 2)}^{3}} + \frac{{(- 8)}^{3}}{\sqrt{- 8 + 44}}$

Step 10.2

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

$\frac{1}{p} \cdot - 2 + \frac{{(- 8)}^{3}}{\sqrt{- 8 + 44}}$

Step 10.3

Combine $\frac{1}{p}$ and $- 2$ .

$\frac{- 2}{p} + \frac{{(- 8)}^{3}}{\sqrt{- 8 + 44}}$

Step 10.4

Move the negative in front of the fraction.

$- \frac{2}{p} + \frac{{(- 8)}^{3}}{\sqrt{- 8 + 44}}$

Step 10.5

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

$- \frac{2}{p} + \frac{- 512}{\sqrt{- 8 + 44}}$

Step 10.6

Simplify the denominator.

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Step 10.6.1

Add $- 8$ and $44$ .

$- \frac{2}{p} + \frac{- 512}{\sqrt{36}}$

Step 10.6.2

Rewrite $36$ as $6^{2}$ .

$- \frac{2}{p} + \frac{- 512}{\sqrt{6^{2}}}$

Step 10.6.3

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

$- \frac{2}{p} + \frac{- 512}{6}$

Step 10.7

Cancel the common factor of $- 512$ and $6$ .

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Step 10.7.1

Factor $2$ out of $- 512$ .

$- \frac{2}{p} + \frac{2 (- 256)}{6}$

Step 10.7.2

Cancel the common factors.

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Step 10.7.2.1

Factor $2$ out of $6$ .

$- \frac{2}{p} + \frac{2 \cdot - 256}{2 \cdot 3}$

Step 10.7.2.2

Cancel the common factor.

$- \frac{2}{p} + \frac{2 \cdot - 256}{2 \cdot 3}$

Step 10.7.2.3

Rewrite the expression.

$- \frac{2}{p} + \frac{- 256}{3}$

Step 10.8

Move the negative in front of the fraction.

$- \frac{2}{p} - \frac{256}{3}$