Calculus Examples

$lim_{x \to 0} x^{\sqrt{x}}$

Step 1

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

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

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $x^{\sqrt{x}}$ as $e^{\ln (x^{\sqrt{x}})}$ .

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

Step 2.2

Expand $\ln (x^{\sqrt{x}})$ by moving $\sqrt{x}$ outside the logarithm.

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

Since the numerator is positive and the denominator $\sqrt{x}$ approaches zero and is greater than zero for $x$ near $0$ to the right, the function increases without bound.

$e^{\frac{- \infty}{\infty}}$

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{- \infty}{\infty}}$

Step 5.2

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.

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

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

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

Step 5.3.2

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

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

Step 5.3.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]}}$

Step 5.3.4

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]}}$

Step 5.3.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 5.3.5.1

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

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [x^{\frac{1}{2} \cdot - 1}]}}$

Step 5.3.5.2

Combine $\frac{1}{2}$ and $- 1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [x^{\frac{- 1}{2}}]}}$

Step 5.3.5.3

Move the negative in front of the fraction.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [x^{- \frac{1}{2}}]}}$

Step 5.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{1}{2}$ .

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

Step 5.3.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3.8

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 5.3.9

Combine the numerators over the common denominator.

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

Step 5.3.10

Simplify the numerator.

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Step 5.3.10.1

Multiply $- 1$ by $2$ .

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

Step 5.3.10.2

Subtract $2$ from $- 1$ .

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

Step 5.3.11

Move the negative in front of the fraction.

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

Step 5.3.12

Simplify.

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Step 5.3.12.1

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

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

Step 5.3.12.2

Combine terms.

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Step 5.3.12.2.1

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

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

Step 5.3.12.2.2

Move $2$ to the left of $x^{\frac{3}{2}}$ .

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

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.5

Combine factors.

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Step 5.5.1

Multiply $2$ by $- 1$ .

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

Step 5.5.2

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

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

Step 5.5.3

Combine $\frac{- 2}{x}$ and $x^{\frac{3}{2}}$ .

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

Step 5.6

Reduce.

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Step 5.6.1

Factor $x$ out of $- 2 x^{\frac{3}{2}}$ .

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

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

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

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

Step 5.6.2.2

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

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

Step 5.6.2.3

Cancel the common factor.

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

Step 5.6.2.4

Rewrite the expression.

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

Step 5.6.2.5

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

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

Step 6

Evaluate the limit.

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Step 6.1

Move the term $- 2$ outside of the limit because it is constant with respect to $x$ .

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

Step 6.2

Move the exponent $\frac{1}{2}$ from $x^{\frac{1}{2}}$ outside the limit using the Limits Power Rule.

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

Step 7

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

$e^{- 2 \cdot 0^{\frac{1}{2}}}$

Step 8

Simplify terms.

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Step 8.1

Simplify the answer.

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Step 8.1.1

Rewrite $0$ as $0^{2}$ .

$e^{- 2 \cdot {(0^{2})}^{\frac{1}{2}}}$

Step 8.1.2

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

$e^{- 2 \cdot 0^{2 (\frac{1}{2})}}$

Step 8.1.3

Cancel the common factor of $2$ .

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Step 8.1.3.1

Cancel the common factor.

$e^{- 2 \cdot 0^{2 (\frac{1}{2})}}$

Step 8.1.3.2

Rewrite the expression.

$e^{- 2 \cdot 0^{1}}$

Step 8.1.4

Evaluate the exponent.

$e^{- 2 \cdot 0}$

Step 8.1.5

Multiply $- 2$ by $0$ .

$e^{0}$

Step 8.2

Anything raised to $0$ is $1$ .

$1$