Calculus Examples

$lim_{x \to - 2} \frac{\sqrt[3]{4 x - x^{3}}}{3 x^{2} \cdot (13 x) + 14}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 2$ .

$\frac{lim_{x \to - 2} \sqrt[3]{4 x - x^{3}}}{lim_{x \to - 2} 3 x^{2} \cdot (13 x) + 14}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt[3]{lim_{x \to - 2} 4 x - x^{3}}}{lim_{x \to - 2} 3 x^{2} \cdot (13 x) + 14}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{\sqrt[3]{lim_{x \to - 2} 4 x - lim_{x \to - 2} x^{3}}}{lim_{x \to - 2} 3 x^{2} \cdot (13 x) + 14}$

Step 4

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

$\frac{\sqrt[3]{4 lim_{x \to - 2} x - lim_{x \to - 2} x^{3}}}{lim_{x \to - 2} 3 x^{2} \cdot (13 x) + 14}$

Step 5

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

$\frac{\sqrt[3]{4 lim_{x \to - 2} x - {(lim_{x \to - 2} x)}^{3}}}{lim_{x \to - 2} 3 x^{2} \cdot (13 x) + 14}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{\sqrt[3]{4 lim_{x \to - 2} x - {(lim_{x \to - 2} x)}^{3}}}{lim_{x \to - 2} 3 x^{2} \cdot 13 x + lim_{x \to - 2} 14}$

Step 7

Move the term $3 \cdot 13$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt[3]{4 lim_{x \to - 2} x - {(lim_{x \to - 2} x)}^{3}}}{3 \cdot 13 lim_{x \to - 2} x^{2} x + lim_{x \to - 2} 14}$

Step 8

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{\sqrt[3]{4 lim_{x \to - 2} x - {(lim_{x \to - 2} x)}^{3}}}{3 \cdot 13 lim_{x \to - 2} x^{2} \cdot lim_{x \to - 2} x + lim_{x \to - 2} 14}$

Step 9

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt[3]{4 lim_{x \to - 2} x - {(lim_{x \to - 2} x)}^{3}}}{3 \cdot 13 {(lim_{x \to - 2} x)}^{2} \cdot lim_{x \to - 2} x + lim_{x \to - 2} 14}$

Step 10

Evaluate the limit of $14$ which is constant as $x$ approaches $- 2$ .

$\frac{\sqrt[3]{4 lim_{x \to - 2} x - {(lim_{x \to - 2} x)}^{3}}}{3 \cdot 13 {(lim_{x \to - 2} x)}^{2} \cdot lim_{x \to - 2} x + 14}$

Step 11

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{\sqrt[3]{4 \cdot - 2 - {(lim_{x \to - 2} x)}^{3}}}{3 \cdot 13 {(lim_{x \to - 2} x)}^{2} \cdot lim_{x \to - 2} x + 14}$

Step 11.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{\sqrt[3]{4 \cdot - 2 - {(- 2)}^{3}}}{3 \cdot 13 {(lim_{x \to - 2} x)}^{2} \cdot lim_{x \to - 2} x + 14}$

Step 11.3

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{\sqrt[3]{4 \cdot - 2 - {(- 2)}^{3}}}{3 \cdot 13 {(- 2)}^{2} \cdot lim_{x \to - 2} x + 14}$

Step 11.4

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{\sqrt[3]{4 \cdot - 2 - {(- 2)}^{3}}}{3 \cdot 13 {(- 2)}^{2} \cdot - 2 + 14}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $4$ by $- 2$ .

$\frac{\sqrt[3]{- 8 - {(- 2)}^{3}}}{3 \cdot 13 {(- 2)}^{2} \cdot - 2 + 14}$

Step 12.1.2

Raise $- 2$ to the power of $3$ .

$\frac{\sqrt[3]{- 8 - - 8}}{3 \cdot 13 {(- 2)}^{2} \cdot - 2 + 14}$

Step 12.1.3

Multiply $- 1$ by $- 8$ .

$\frac{\sqrt[3]{- 8 + 8}}{3 \cdot 13 {(- 2)}^{2} \cdot - 2 + 14}$

Step 12.1.4

Add $- 8$ and $8$ .

$\frac{\sqrt[3]{0}}{3 \cdot 13 {(- 2)}^{2} \cdot - 2 + 14}$

Step 12.1.5

Rewrite $0$ as $0^{3}$ .

$\frac{\sqrt[3]{0^{3}}}{3 \cdot 13 {(- 2)}^{2} \cdot - 2 + 14}$

Step 12.1.6

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

$\frac{0}{3 \cdot 13 {(- 2)}^{2} \cdot - 2 + 14}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply ${(- 2)}^{2}$ by $- 2$ by adding the exponents.

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Step 12.2.1.1

Move $- 2$ .

$\frac{0}{3 \cdot 13 (- 2 {(- 2)}^{2}) + 14}$

Step 12.2.1.2

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 12.2.1.2.1

Raise $- 2$ to the power of $1$ .

$\frac{0}{3 \cdot 13 ({(- 2)}^{1} {(- 2)}^{2}) + 14}$

Step 12.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{0}{3 \cdot 13 {(- 2)}^{1 + 2} + 14}$

Step 12.2.1.3

Add $1$ and $2$ .

$\frac{0}{3 \cdot 13 {(- 2)}^{3} + 14}$

Step 12.2.2

Multiply $3$ by $13$ .

$\frac{0}{39 {(- 2)}^{3} + 14}$

Step 12.2.3

Raise $- 2$ to the power of $3$ .

$\frac{0}{39 \cdot - 8 + 14}$

Step 12.2.4

Multiply $39$ by $- 8$ .

$\frac{0}{- 312 + 14}$

Step 12.2.5

Add $- 312$ and $14$ .

$\frac{0}{- 298}$

Step 12.3

Divide $0$ by $- 298$ .

$0$