Calculus Examples

$lim_{x \to - 3} \sqrt[3]{\frac{x - 4}{6 \cdot x^{2} + 2}}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to - 3} \frac{x - 4}{6 \cdot x^{2} + 2}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 3$ .

$\sqrt[3]{\frac{lim_{x \to - 3} x - 4}{lim_{x \to - 3} 6 \cdot x^{2} + 2}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\sqrt[3]{\frac{lim_{x \to - 3} x - lim_{x \to - 3} 4}{lim_{x \to - 3} 6 \cdot x^{2} + 2}}$

Step 4

Evaluate the limit of $4$ which is constant as $x$ approaches $- 3$ .

$\sqrt[3]{\frac{lim_{x \to - 3} x - 1 \cdot 4}{lim_{x \to - 3} 6 \cdot x^{2} + 2}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\sqrt[3]{\frac{lim_{x \to - 3} x - 1 \cdot 4}{lim_{x \to - 3} 6 \cdot x^{2} + lim_{x \to - 3} 2}}$

Step 6

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{lim_{x \to - 3} x - 1 \cdot 4}{6 lim_{x \to - 3} x^{2} + lim_{x \to - 3} 2}}$

Step 7

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

$\sqrt[3]{\frac{lim_{x \to - 3} x - 1 \cdot 4}{6 {(lim_{x \to - 3} x)}^{2} + lim_{x \to - 3} 2}}$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $- 3$ .

$\sqrt[3]{\frac{lim_{x \to - 3} x - 1 \cdot 4}{6 {(lim_{x \to - 3} x)}^{2} + 2}}$

Step 9

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\sqrt[3]{\frac{- 3 - 1 \cdot 4}{6 {(lim_{x \to - 3} x)}^{2} + 2}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\sqrt[3]{\frac{- 3 - 1 \cdot 4}{6 {(- 3)}^{2} + 2}}$

Step 10

Simplify the answer.

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Step 10.1

Multiply $- 1$ by $4$ .

$\sqrt[3]{\frac{- 3 - 4}{6 {(- 3)}^{2} + 2}}$

Step 10.2

Subtract $4$ from $- 3$ .

$\sqrt[3]{\frac{- 7}{6 {(- 3)}^{2} + 2}}$

Step 10.3

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

$\sqrt[3]{\frac{- 7}{6 \cdot 9 + 2}}$

Step 10.4

Multiply $6$ by $9$ .

$\sqrt[3]{\frac{- 7}{54 + 2}}$

Step 10.5

Add $54$ and $2$ .

$\sqrt[3]{\frac{- 7}{56}}$

Step 10.6

Reduce the expression by cancelling the common factors.

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Step 10.6.1

Cancel the common factor of $- 7$ and $56$ .

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Step 10.6.1.1

Factor $7$ out of $- 7$ .

$\sqrt[3]{\frac{7 (- 1)}{56}}$

Step 10.6.1.2

Cancel the common factors.

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Step 10.6.1.2.1

Factor $7$ out of $56$ .

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

Step 10.6.1.2.2

Cancel the common factor.

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

Step 10.6.1.2.3

Rewrite the expression.

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

Step 10.6.2

Move the negative in front of the fraction.

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

Step 10.7

Rewrite $- \frac{1}{8}$ as ${(- \frac{1}{2})}^{3}$ .

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

Step 10.8

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

$- \frac{1}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$