Calculus Examples

$lim_{x \to 0} \frac{x^{\frac{1}{3}}}{x}$

Step 1

Reduce.

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Step 1.1

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

$lim_{x \to 0} \frac{x \cdot x^{\frac{- 2}{3}}}{x}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to 0} \frac{x \cdot x^{\frac{- 2}{3}}}{x^{1}}$

Step 1.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to 0} \frac{x \cdot x^{\frac{- 2}{3}}}{x \cdot 1}$

Step 1.2.3

Cancel the common factor.

$lim_{x \to 0} \frac{x \cdot x^{\frac{- 2}{3}}}{x \cdot 1}$

Step 1.2.4

Rewrite the expression.

$lim_{x \to 0} \frac{x^{\frac{- 2}{3}}}{1}$

Step 1.2.5

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

$lim_{x \to 0} x^{\frac{- 2}{3}}$

Step 2

Move the exponent $\frac{- 2}{3}$ from $x^{\frac{- 2}{3}}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 0} x)}^{\frac{- 2}{3}}$

Step 3

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

$0^{\frac{- 2}{3}}$

Step 4

Simplify the answer.

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Step 4.1

Move the negative in front of the fraction.

$0^{- \frac{2}{3}}$

Step 4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{0^{\frac{2}{3}}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $0$ as $0^{3}$ .

$\frac{1}{{(0^{3})}^{\frac{2}{3}}}$

Step 4.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{0^{3 (\frac{2}{3})}}$

Step 4.3.3

Cancel the common factor of $3$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{1}{0^{3 (\frac{2}{3})}}$

Step 4.3.3.2

Rewrite the expression.

$\frac{1}{0^{2}}$

Step 4.3.4

Raising $0$ to any positive power yields $0$ .

$\frac{1}{0}$

Step 4.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{0}$

Step 4.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{0}$

Step 5

The expression contains a division by $0$ . The expression is undefined.

Undefined