Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

Expand $\ln ({(\frac{x}{2})}^{\frac{1}{x - 2}})$ by moving $\frac{1}{x - 2}$ outside the logarithm.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

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

$e^{lim_{x \to 2} \frac{\ln (\frac{x}{2})}{x - 2}}$

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.

$e^{\frac{lim_{x \to 2} \ln (\frac{x}{2})}{lim_{x \to 2} x - 2}}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

Move the limit inside the logarithm.

$e^{\frac{\ln (lim_{x \to 2} \frac{x}{2})}{lim_{x \to 2} x - 2}}$

Step 3.1.2.1.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.1.2.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$e^{\frac{\ln (\frac{1}{2} \cdot 2)}{lim_{x \to 2} x - 2}}$

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Cancel the common factor of $2$ .

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Step 3.1.2.3.1.1

Cancel the common factor.

$e^{\frac{\ln (\frac{1}{2} \cdot 2)}{lim_{x \to 2} x - 2}}$

Step 3.1.2.3.1.2

Rewrite the expression.

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

Step 3.1.2.3.2

The natural logarithm of $1$ is $0$ .

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 3.1.3.1.2

Evaluate the limit of $2$ which is constant as $x$ approaches $2$ .

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

Step 3.1.3.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Multiply $- 1$ by $2$ .

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

Step 3.1.3.3.2

Subtract $2$ from $2$ .

$e^{\frac{0}{0}}$

Step 3.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{0}{0}}$

Step 3.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{0}{0}}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{0}{0}}$

Step 3.2

Since $\frac{0}{0}$ 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.

$lim_{x \to 2} \frac{\ln (\frac{x}{2})}{x - 2} = lim_{x \to 2} \frac{\frac{d}{d x} [\ln (\frac{x}{2})]}{\frac{d}{d x} [x - 2]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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 $g (x) = \frac{x}{2}$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$e^{lim_{x \to 2} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{x}{2}]}{\frac{d}{d x} [x - 2]}}$

Step 3.3.2.2

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

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

Step 3.3.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

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

Step 3.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{x}{2}$ .

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

Step 3.3.4

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

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

Step 3.3.5

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 3.3.6

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

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

Step 3.3.7

Cancel the common factor of $2$ .

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Step 3.3.7.1

Cancel the common factor.

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

Step 3.3.7.2

Rewrite the expression.

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

Step 3.3.8

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

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

Step 3.3.9

Multiply $\frac{1}{x}$ by $1$ .

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

Step 3.3.10

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

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

Step 3.3.11

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

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

Step 3.3.12

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 3.3.13

Add $1$ and $0$ .

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Multiply $\frac{1}{x}$ by $1$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

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

Step 4.2

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

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

Step 5

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$e^{\frac{1}{2}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{1}{2}}$

Decimal Form:

$1.64872127 \dots$