Calculus Examples

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

Step 1

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

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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 2.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

The derivative of $\log (x)$ with respect to $x$ is $\frac{1}{x \ln (10)}$ .

$lim_{x \to 0^{+}} \frac{\frac{1}{x \ln (10)}}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.3

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

$lim_{x \to 0^{+}} \frac{\frac{1}{x \ln (10)}}{\frac{d}{d x} [x^{- 1}]}$

Step 2.3.4

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

$lim_{x \to 0^{+}} \frac{\frac{1}{x \ln (10)}}{- x^{- 2}}$

Step 2.3.5

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

$lim_{x \to 0^{+}} \frac{\frac{1}{x \ln (10)}}{- \frac{1}{x^{2}}}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0^{+}} \frac{1}{x \ln (10)} (- x^{2})$

Step 2.5

Combine $\frac{1}{x \ln (10)}$ and $x^{2}$ .

$lim_{x \to 0^{+}} - \frac{x^{2}}{x \ln (10)}$

Step 2.6

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

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Step 2.6.1

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

$lim_{x \to 0^{+}} - \frac{x \cdot x}{x \ln (10)}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $x$ out of $x \ln (10)$ .

$lim_{x \to 0^{+}} - \frac{x \cdot x}{x (\ln (10))}$

Step 2.6.2.2

Cancel the common factor.

$lim_{x \to 0^{+}} - \frac{x \cdot x}{x \ln (10)}$

Step 2.6.2.3

Rewrite the expression.

$lim_{x \to 0^{+}} - \frac{x}{\ln (10)}$

Step 3

Evaluate the limit.

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Step 3.1

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

$- lim_{x \to 0^{+}} \frac{x}{\ln (10)}$

Step 3.2

Move the term $\frac{1}{\ln (10)}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{\ln (10)} lim_{x \to 0^{+}} x$

Step 4

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

$- \frac{1}{\ln (10)} \cdot 0$

Step 5

Multiply $- \frac{1}{\ln (10)} \cdot 0$ .

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Step 5.1

Multiply $0$ by $- 1$ .

$0 \frac{1}{\ln (10)}$

Step 5.2

Multiply $0$ by $\frac{1}{\ln (10)}$ .

$0$