Calculus Examples

$lim_{x \to 7} \sqrt[3]{\frac{2 - 8 x}{x - 5}}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to 7} \frac{2 - 8 x}{x - 5}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $7$ .

$\sqrt[3]{\frac{lim_{x \to 7} 2 - 8 x}{lim_{x \to 7} x - 5}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\sqrt[3]{\frac{lim_{x \to 7} 2 - lim_{x \to 7} 8 x}{lim_{x \to 7} x - 5}}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $7$ .

$\sqrt[3]{\frac{2 - lim_{x \to 7} 8 x}{lim_{x \to 7} x - 5}}$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{2 - 8 lim_{x \to 7} x}{lim_{x \to 7} x - 5}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\sqrt[3]{\frac{2 - 8 lim_{x \to 7} x}{lim_{x \to 7} x - lim_{x \to 7} 5}}$

Step 7

Evaluate the limit of $5$ which is constant as $x$ approaches $7$ .

$\sqrt[3]{\frac{2 - 8 lim_{x \to 7} x}{lim_{x \to 7} x - 1 \cdot 5}}$

Step 8

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\sqrt[3]{\frac{2 - 8 \cdot 7}{lim_{x \to 7} x - 1 \cdot 5}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\sqrt[3]{\frac{2 - 8 \cdot 7}{7 - 1 \cdot 5}}$

Step 9

Simplify the answer.

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Step 9.1

Multiply $- 8$ by $7$ .

$\sqrt[3]{\frac{2 - 56}{7 - 1 \cdot 5}}$

Step 9.2

Subtract $56$ from $2$ .

$\sqrt[3]{\frac{- 54}{7 - 1 \cdot 5}}$

Step 9.3

Multiply $- 1$ by $5$ .

$\sqrt[3]{\frac{- 54}{7 - 5}}$

Step 9.4

Subtract $5$ from $7$ .

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

Step 9.5

Divide $- 54$ by $2$ .

$\sqrt[3]{- 27}$

Step 9.6

Rewrite $- 27$ as ${(- 3)}^{3}$ .

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

Step 9.7

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

$- 3$