Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the limit under the radical sign.

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

Step 4

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

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

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 8

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

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

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

$\frac{\sqrt[3]{1 - 2 \cdot 3} + 1}{3 - 1 \cdot 1}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $- 2$ by $3$ .

$\frac{\sqrt[3]{1 - 6} + 1}{3 - 1 \cdot 1}$

Step 11.1.2

Subtract $6$ from $1$ .

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

Step 11.1.3

Rewrite $- 5$ as ${(- 1)}^{3} \cdot 5$ .

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Step 11.1.3.1

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{\sqrt[3]{- 1 (5)} + 1}{3 - 1 \cdot 1}$

Step 11.1.3.2

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

$\frac{\sqrt[3]{{(- 1)}^{3} \cdot 5} + 1}{3 - 1 \cdot 1}$

Step 11.1.4

Pull terms out from under the radical.

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

Step 11.1.5

Rewrite $- 1 \sqrt[3]{5}$ as $- \sqrt[3]{5}$ .

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

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $- 1$ by $1$ .

$\frac{- \sqrt[3]{5} + 1}{3 - 1}$

Step 11.2.2

Subtract $1$ from $3$ .

$\frac{- \sqrt[3]{5} + 1}{2}$

Step 11.3

Factor $- 1$ out of $- \sqrt[3]{5}$ .

$\frac{- (\sqrt[3]{5}) + 1}{2}$

Step 11.4

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (\sqrt[3]{5}) - 1 (- 1)}{2}$

Step 11.5

Factor $- 1$ out of $- (\sqrt[3]{5}) - 1 (- 1)$ .

$\frac{- (\sqrt[3]{5} - 1)}{2}$

Step 11.6

Rewrite $- (\sqrt[3]{5} - 1)$ as $- 1 (\sqrt[3]{5} - 1)$ .

$\frac{- 1 (\sqrt[3]{5} - 1)}{2}$

Step 11.7

Move the negative in front of the fraction.

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.35498797 \dots$