Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite

${(\frac{1}{x})}^{x}$ as

$e^{\ln ({(\frac{1}{x})}^{x})}$ .

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

Step 1.2

Expand

$\ln ({(\frac{1}{x})}^{x})$ by moving

$x$ outside the logarithm.

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

Step 2

Move the limit into the exponent.

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

Step 3

Rewrite

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

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

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

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

As log approaches infinity, the value goes to

$\infty$ .

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

Step 4.1.3

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.

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

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 4.2

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \ln (x)$ and

$g (x) = \frac{1}{x}$ .

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Step 4.3.2.1

To apply the Chain Rule, set

$u$ as

$\frac{1}{x}$ .

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

Step 4.3.2.2

The derivative of

$\ln (u)$ with respect to

$u$ is

$\frac{1}{u}$ .

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

Step 4.3.2.3

Replace all occurrences of

$u$ with

$\frac{1}{x}$ .

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

Step 4.3.3

Multiply by the reciprocal of the fraction to divide by

$\frac{1}{x}$ .

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

Step 4.3.4

Multiply

$x$ by

$1$ .

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

Step 4.3.5

Rewrite

$\frac{1}{x}$ as

$x^{- 1}$ .

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

Step 4.3.6

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = - 1$ .

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

Step 4.3.7

Raise

$x$ to the power of

$1$ .

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

Step 4.3.8

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3.9

Subtract

$2$ from

$1$ .

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

Step 4.3.10

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.3.11

Rewrite

$\frac{1}{x}$ as

$x^{- 1}$ .

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

Step 4.3.12

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = - 1$ .

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

Step 4.3.13

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.5

Combine factors.

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Step 4.5.1

Multiply

$- 1$ by

$- 1$ .

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

Step 4.5.2

Multiply

$\frac{1}{x}$ by

$1$ .

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

Step 4.5.3

Combine

$\frac{1}{x}$ and

$x^{2}$ .

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

Step 4.6

Cancel the common factor of

$x^{2}$ and

$x$ .

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Step 4.6.1

Factor

$x$ out of

$x^{2}$ .

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

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Raise

$x$ to the power of

$1$ .

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

Step 4.6.2.2

Factor

$x$ out of

$x^{1}$ .

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

Step 4.6.2.3

Cancel the common factor.

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

Step 4.6.2.4

Rewrite the expression.

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

Step 4.6.2.5

Divide

$x$ by

$1$ .

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

Step 5

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$e^{0}$

Step 6

Anything raised to

$0$ is

$1$ .

$1$