Calculus Examples

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

Step 1

Reduce.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 1}$ as ${(x - 1)}^{\frac{1}{3}}$ .

$lim_{x \to 1} \frac{{(x - 1)}^{\frac{1}{3}}}{x - 1}$

Step 1.2

Factor $x - 1$ out of ${(x - 1)}^{\frac{1}{3}}$ .

$lim_{x \to 1} \frac{(x - 1) {(x - 1)}^{\frac{- 2}{3}}}{x - 1}$

Step 1.3

Cancel the common factors.

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Step 1.3.1

Multiply by $1$ .

$lim_{x \to 1} \frac{(x - 1) {(x - 1)}^{\frac{- 2}{3}}}{(x - 1) \cdot 1}$

Step 1.3.2

Cancel the common factor.

$lim_{x \to 1} \frac{(x - 1) {(x - 1)}^{\frac{- 2}{3}}}{(x - 1) \cdot 1}$

Step 1.3.3

Rewrite the expression.

$lim_{x \to 1} \frac{{(x - 1)}^{\frac{- 2}{3}}}{1}$

Step 1.3.4

Divide ${(x - 1)}^{\frac{- 2}{3}}$ by $1$ .

$lim_{x \to 1} {(x - 1)}^{\frac{- 2}{3}}$

Step 1.4

Move ${(x - 1)}^{\frac{- 2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 1} \frac{1}{{(x - 1)}^{- \frac{- 2}{3}}}$

Step 2

Since the function approaches $- \infty$ from the left and $\infty$ from the right, the limit does not exist.

$Does not exist$