Calculus Examples

$lim_{x \to 0} x^{3} \ln (x)$

Step 1

Change the two-sided limit into a right sided limit.

$lim_{x \to 0^{+}} x^{3} \ln (x)$

Step 2

Rewrite $x^{3} \ln (x)$ as $\frac{\ln (x)}{x^{- 3}}$ .

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 3.1.2

As $x$ approaches $0$ from the right side, $\ln (x)$ decreases without bound.

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

Since the numerator is a constant and the denominator approaches $0$ when $x$ approaches $0$ from the right, the fraction $\frac{1}{x^{3}}$ approaches infinity.

$\frac{- \infty}{\infty}$

Step 3.1.3.3

Infinity divided by infinity is undefined.

Undefined

$\frac{- \infty}{\infty}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{- \infty}{\infty}$

Step 3.2

Since $\frac{- \infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0^{+}} \frac{\ln (x)}{x^{- 3}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [x^{- 3}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [x^{- 3}]}$

Step 3.3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [x^{- 3}]}$

Step 3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$lim_{x \to 0^{+}} \frac{\frac{1}{x}}{- 3 x^{- 4}}$

Step 3.3.4

Simplify.

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Step 3.3.4.1

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

$lim_{x \to 0^{+}} \frac{\frac{1}{x}}{- 3 \frac{1}{x^{4}}}$

Step 3.3.4.2

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- 3}{x^{4}}}$

Step 3.3.4.3

Move the negative in front of the fraction.

$lim_{x \to 0^{+}} \frac{\frac{1}{x}}{- \frac{3}{x^{4}}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0^{+}} \frac{1}{x} (- \frac{x^{4}}{3})$

Step 3.5

Multiply $\frac{1}{x}$ by $\frac{x^{4}}{3}$ .

$lim_{x \to 0^{+}} - \frac{x^{4}}{x \cdot 3}$

Step 3.6

Cancel the common factor of $x^{4}$ and $x$ .

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Step 3.6.1

Factor $x$ out of $x^{4}$ .

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

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $x$ out of $x \cdot 3$ .

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

Step 3.6.2.2

Cancel the common factor.

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

Step 3.6.2.3

Rewrite the expression.

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

Step 4

Evaluate the limit.

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Step 4.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 4.2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

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

Step 4.3

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

$- \frac{1}{3} {(lim_{x \to 0^{+}} x)}^{3}$

Step 5

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

$- \frac{1}{3} \cdot 0^{3}$

Step 6

Simplify the answer.

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Step 6.1

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

$- \frac{1}{3} \cdot 0$

Step 6.2

Multiply $- \frac{1}{3} \cdot 0$ .

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Step 6.2.1

Multiply $0$ by $- 1$ .

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

Step 6.2.2

Multiply $0$ by $\frac{1}{3}$ .

$0$