Calculus Examples

$lim_{x \to 0} \sqrt[3]{\frac{3}{2 x - 1}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to 0} \frac{3}{2 x - 1}}$

Step 1.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{3 lim_{x \to 0} \frac{1}{2 x - 1}}$

Step 1.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\sqrt[3]{3 \frac{lim_{x \to 0} 1}{lim_{x \to 0} 2 x - 1}}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\sqrt[3]{3 \frac{1}{lim_{x \to 0} 2 x - 1}}$

Step 1.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\sqrt[3]{3 \frac{1}{lim_{x \to 0} 2 x - lim_{x \to 0} 1}}$

Step 1.6

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

$\sqrt[3]{3 \frac{1}{2 lim_{x \to 0} x - lim_{x \to 0} 1}}$

Step 1.7

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\sqrt[3]{3 \frac{1}{2 lim_{x \to 0} x - 1 \cdot 1}}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sqrt[3]{3 \frac{1}{2 \cdot 0 - 1 \cdot 1}}$

Step 3

Simplify the answer.

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Step 3.1

Combine $3$ and $\frac{1}{2 \cdot 0 - 1 \cdot 1}$ .

$\sqrt[3]{\frac{3}{2 \cdot 0 - 1}}$

Step 3.2

Multiply $2$ by $0$ .

$\sqrt[3]{\frac{3}{0 - 1}}$

Step 3.3

Subtract $1$ from $0$ .

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

Step 3.4

Divide $3$ by $- 1$ .

$\sqrt[3]{- 3}$

Step 3.5

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

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Step 3.5.1

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 3.5.2

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

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

Step 3.6

Pull terms out from under the radical.

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

Step 3.7

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

$- \sqrt[3]{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \sqrt[3]{3}$

Decimal Form:

$- 1.44224957 \dots$