Calculus Examples

$lim_{x \to 4} \sqrt[3]{\frac{x}{- 7 x + 1}}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to 4} \frac{x}{- 7 x + 1}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\sqrt[3]{\frac{lim_{x \to 4} x}{lim_{x \to 4} - 7 x + 1}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\sqrt[3]{\frac{lim_{x \to 4} x}{- lim_{x \to 4} 7 x + lim_{x \to 4} 1}}$

Step 4

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

$\sqrt[3]{\frac{lim_{x \to 4} x}{- 7 lim_{x \to 4} x + lim_{x \to 4} 1}}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\sqrt[3]{\frac{lim_{x \to 4} x}{- 7 lim_{x \to 4} x + 1}}$

Step 6

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\sqrt[3]{\frac{4}{- 7 lim_{x \to 4} x + 1}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\sqrt[3]{\frac{4}{- 7 \cdot 4 + 1}}$

Step 7

Simplify the answer.

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Step 7.1

Multiply $- 7$ by $4$ .

$\sqrt[3]{\frac{4}{- 28 + 1}}$

Step 7.2

Add $- 28$ and $1$ .

$\sqrt[3]{\frac{4}{- 27}}$

Step 7.3

Move the negative in front of the fraction.

$\sqrt[3]{- \frac{4}{27}}$

Step 7.4

Rewrite $- \frac{4}{27}$ as ${(- \frac{1}{3})}^{3} \cdot 4$ .

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Step 7.4.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{4}{27}}$

Step 7.4.2

Factor the perfect power $1^{3}$ out of $4$ .

$\sqrt[3]{{(- 1)}^{3} \frac{1^{3} \cdot 4}{27}}$

Step 7.4.3

Factor the perfect power $3^{3}$ out of $27$ .

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

Step 7.4.4

Rearrange the fraction $\frac{1^{3} \cdot 4}{3^{3} \cdot 1}$ .

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

Step 7.4.5

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

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

Step 7.5

Pull terms out from under the radical.

$- \frac{1}{3} \sqrt[3]{4}$

Step 7.6

Combine $\sqrt[3]{4}$ and $\frac{1}{3}$ .

$- \frac{\sqrt[3]{4}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt[3]{4}}{3}$

Decimal Form:

$- 0.52913368 \dots$