Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Move the limit inside the logarithm.

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

Step 1.2.2

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

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

Step 1.2.3

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

$\frac{0}{lim_{x \to 1} x - 1}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x - lim_{x \to 1} 1}$

Step 1.3.1.2

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

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

Step 1.3.2

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

$\frac{0}{1 - 1 \cdot 1}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $0$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{1}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 1} \frac{1}{x} \cdot 1$

Step 5

Evaluate the limit.

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Step 5.1

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

$lim_{x \to 1} \frac{1}{x}$

Step 5.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 1}{lim_{x \to 1} x}$

Step 5.3

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

$\frac{1}{lim_{x \to 1} x}$

Step 6

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

$\frac{1}{1}$

Step 7

Divide $1$ by $1$ .

$1$