Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

Tap for more steps...

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.

Tap for more steps...

Step 4.1

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

Tap for more steps...

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.

Tap for more steps...

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

Tap for more steps...

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.

Tap for more steps...

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 $x$ .

Tap for more steps...

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.

Tap for more steps...

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 $1$ .

$1$