Calculus Examples

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

Step 1

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

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

Step 2

Rewrite $x \ln (x)$ as $\frac{\ln (x)}{\frac{1}{x}}$ .

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

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.

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

Step 3.1.2

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

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

Step 3.1.3

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

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

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.

$lim_{x \to 0^{+}} \frac{\ln (x)}{\frac{1}{x}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [\frac{1}{x}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Combine $\frac{1}{x}$ and $x^{2}$ .

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

Step 3.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 3.6.1

Factor $x$ out of $x^{2}$ .

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

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

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

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

Step 3.6.2.2

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

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

Step 3.6.2.3

Cancel the common factor.

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

Step 3.6.2.4

Rewrite the expression.

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

Step 3.6.2.5

Divide $x$ by $1$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Evaluate the limit.

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Step 4.1.1

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

$1 lim_{x \to 0^{+}} x$

Step 4.1.2

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

$- 0$

Step 4.2

Multiply $- 1$ by $0$ .

$0$