Calculus Examples

$lim_{x \to - 8} \frac{1 + 7 \sqrt[3]{x}}{2 \sqrt[3]{x}}$

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to - 8} \frac{1 + 7 \sqrt[3]{x}}{\sqrt[3]{x}}$

Step 2

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

$\frac{1}{2} \cdot \frac{lim_{x \to - 8} 1 + 7 \sqrt[3]{x}}{lim_{x \to - 8} \sqrt[3]{x}}$

Step 3

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

$\frac{1}{2} \cdot \frac{lim_{x \to - 8} 1 + lim_{x \to - 8} 7 \sqrt[3]{x}}{lim_{x \to - 8} \sqrt[3]{x}}$

Step 4

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

$\frac{1}{2} \cdot \frac{1 + lim_{x \to - 8} 7 \sqrt[3]{x}}{lim_{x \to - 8} \sqrt[3]{x}}$

Step 5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{1 + 7 lim_{x \to - 8} \sqrt[3]{x}}{lim_{x \to - 8} \sqrt[3]{x}}$

Step 6

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{1 + 7 \sqrt[3]{lim_{x \to - 8} x}}{lim_{x \to - 8} \sqrt[3]{x}}$

Step 7

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{1 + 7 \sqrt[3]{lim_{x \to - 8} x}}{\sqrt[3]{lim_{x \to - 8} x}}$

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 $x$ .

$\frac{1}{2} \cdot \frac{1 + 7 \sqrt[3]{- 8}}{\sqrt[3]{lim_{x \to - 8} x}}$

Step 8.2

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

$\frac{1}{2} \cdot \frac{1 + 7 \sqrt[3]{- 8}}{\sqrt[3]{- 8}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{1}{2} \cdot \frac{1 + 7 \sqrt[3]{{(- 2)}^{3}}}{\sqrt[3]{- 8}}$

Step 9.1.2

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

$\frac{1}{2} \cdot \frac{1 + 7 \cdot - 2}{\sqrt[3]{- 8}}$

Step 9.1.3

Multiply $7$ by $- 2$ .

$\frac{1}{2} \cdot \frac{1 - 14}{\sqrt[3]{- 8}}$

Step 9.1.4

Subtract $14$ from $1$ .

$\frac{1}{2} \cdot \frac{- 13}{\sqrt[3]{- 8}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$\frac{1}{2} \cdot \frac{- 13}{\sqrt[3]{{(- 2)}^{3}}}$

Step 9.2.2

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

$\frac{1}{2} \cdot \frac{- 13}{- 2}$

Step 9.3

Dividing two negative values results in a positive value.

$\frac{1}{2} \cdot \frac{13}{2}$

Step 9.4

Multiply $\frac{1}{2} \cdot \frac{13}{2}$ .

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Step 9.4.1

Multiply $\frac{1}{2}$ by $\frac{13}{2}$ .

$\frac{13}{2 \cdot 2}$

Step 9.4.2

Multiply $2$ by $2$ .

$\frac{13}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{4}$

Decimal Form:

$3.25$