Calculus Examples

$lim_{x \to 5} \frac{\sqrt[3]{58 x - 7}}{4 - x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} \sqrt[3]{58 x - 7}}{lim_{x \to 5} 4 - x}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt[3]{lim_{x \to 5} 58 x - 7}}{lim_{x \to 5} 4 - x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sqrt[3]{lim_{x \to 5} 58 x - lim_{x \to 5} 7}}{lim_{x \to 5} 4 - x}$

Step 4

Move the term $58$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt[3]{58 lim_{x \to 5} x - lim_{x \to 5} 7}}{lim_{x \to 5} 4 - x}$

Step 5

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

$\frac{\sqrt[3]{58 lim_{x \to 5} x - 1 \cdot 7}}{lim_{x \to 5} 4 - x}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sqrt[3]{58 lim_{x \to 5} x - 1 \cdot 7}}{lim_{x \to 5} 4 - lim_{x \to 5} x}$

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $5$ .

$\frac{\sqrt[3]{58 lim_{x \to 5} x - 1 \cdot 7}}{4 - lim_{x \to 5} x}$

Step 8

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sqrt[3]{58 \cdot 5 - 1 \cdot 7}}{4 - lim_{x \to 5} x}$

Step 8.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sqrt[3]{58 \cdot 5 - 1 \cdot 7}}{4 - 5}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $58$ by $5$ .

$\frac{\sqrt[3]{290 - 1 \cdot 7}}{4 - 5}$

Step 9.1.2

Multiply $- 1$ by $7$ .

$\frac{\sqrt[3]{290 - 7}}{4 - 5}$

Step 9.1.3

Subtract $7$ from $290$ .

$\frac{\sqrt[3]{283}}{4 - 5}$

Step 9.2

Subtract $5$ from $4$ .

$\frac{\sqrt[3]{283}}{- 1}$

Step 9.3

Move the negative one from the denominator of $\frac{\sqrt[3]{283}}{- 1}$ .

$- 1 \cdot \sqrt[3]{283}$

Step 9.4

Rewrite $- 1 \cdot \sqrt[3]{283}$ as $- \sqrt[3]{283}$ .

$- \sqrt[3]{283}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \sqrt[3]{283}$

Decimal Form:

$- 6.56541442 \dots$