Calculus Examples

$lim_{x \to 1} x^{\frac{6}{x - 1}}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite $x^{\frac{6}{x - 1}}$ as $e^{\ln (x^{\frac{6}{x - 1}})}$ .

$lim_{x \to 1} e^{\ln (x^{\frac{6}{x - 1}})}$

Step 1.2

Expand $\ln (x^{\frac{6}{x - 1}})$ by moving $\frac{6}{x - 1}$ outside the logarithm.

$lim_{x \to 1} e^{\frac{6}{x - 1} \ln (x)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 1} \frac{6}{x - 1} \ln (x)}$

Step 2.2

Combine $\frac{6}{x - 1}$ and $\ln (x)$ .

$e^{lim_{x \to 1} \frac{6 \ln (x)}{x - 1}}$

Step 2.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$e^{6 lim_{x \to 1} \frac{\ln (x)}{x - 1}}$

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.

$e^{6 \frac{lim_{x \to 1} \ln (x)}{lim_{x \to 1} x - 1}}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Move the limit inside the logarithm.

$e^{6 \frac{\ln (lim_{x \to 1} x)}{lim_{x \to 1} x - 1}}$

Step 3.1.2.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$e^{6 \frac{\ln (1)}{lim_{x \to 1} x - 1}}$

Step 3.1.2.3

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

$e^{6 \frac{0}{lim_{x \to 1} x - 1}}$

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 $1$ .

$e^{6 \frac{0}{lim_{x \to 1} x - lim_{x \to 1} 1}}$

Step 3.1.3.1.2

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

$e^{6 \frac{0}{lim_{x \to 1} x - 1 \cdot 1}}$

Step 3.1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$e^{6 \frac{0}{1 - 1 \cdot 1}}$

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Multiply $- 1$ by $1$ .

$e^{6 \frac{0}{1 - 1}}$

Step 3.1.3.3.2

Subtract $1$ from $1$ .

$e^{6 (\frac{0}{0})}$

Step 3.1.3.3.3

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

Undefined

$e^{6 (\frac{0}{0})}$

Step 3.1.3.4

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

Undefined

$e^{6 (\frac{0}{0})}$

Step 3.1.4

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

Undefined

$e^{6 (\frac{0}{0})}$

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$e^{6 lim_{x \to 1} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [x - 1]}}$

Step 3.3.2

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

$e^{6 lim_{x \to 1} \frac{\frac{1}{x}}{\frac{d}{d x} [x - 1]}}$

Step 3.3.3

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$e^{6 lim_{x \to 1} \frac{\frac{1}{x}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 1]}}$

Step 3.3.4

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

$e^{6 lim_{x \to 1} \frac{\frac{1}{x}}{1 + \frac{d}{d x} [- 1]}}$

Step 3.3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$e^{6 lim_{x \to 1} \frac{\frac{1}{x}}{1 + 0}}$

Step 3.3.6

Add $1$ and $0$ .

$e^{6 lim_{x \to 1} \frac{\frac{1}{x}}{1}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{6 lim_{x \to 1} \frac{1}{x} \cdot 1}$

Step 3.5

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

$e^{6 lim_{x \to 1} \frac{1}{x}}$

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 $1$ .

$e^{6 \frac{lim_{x \to 1} 1}{lim_{x \to 1} x}}$

Step 4.2

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

$e^{6 \frac{1}{lim_{x \to 1} x}}$

Step 5

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$e^{6 (\frac{1}{1})}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $1$ .

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Step 6.1.1

Cancel the common factor.

$e^{6 (\frac{1}{1})}$

Step 6.1.2

Rewrite the expression.

$e^{6 \cdot 1}$

Step 6.2

Multiply $6$ by $1$ .

$e^{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$e^{6}$

Decimal Form:

$403.42879349 \dots$