Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} x - 1 - \ln (x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 1} x - lim_{x \to 1} 1 - lim_{x \to 1} \ln (x)}$

Step 1.1.3.2

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

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

Step 1.1.3.3

Move the limit inside the logarithm.

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

Step 1.1.3.4

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 1.1.3.4.1

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

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

Step 1.1.3.4.2

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

$\frac{0}{1 - 1 \cdot 1 - \ln (1)}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1 - \ln (1)}$

Step 1.1.3.5.1.2

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

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

Step 1.1.3.5.1.3

Multiply $- 1$ by $0$ .

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

Step 1.1.3.5.2

Subtract $1$ from $1$ .

$\frac{0}{0 + 0}$

Step 1.1.3.5.3

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $1$ and $0$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

Evaluate $\frac{d}{d x} [- \ln (x)]$ .

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Step 1.3.9.1

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

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

Step 1.3.9.2

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

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

Step 1.3.10

Simplify.

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Step 1.3.10.1

Add $1$ and $0$ .

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

Step 1.3.10.2

Reorder terms.

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

Step 1.4

Combine terms.

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Step 1.4.1

Write $1$ as a fraction with a common denominator.

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

Step 1.4.2

Combine the numerators over the common denominator.

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

Step 2

Since the function approaches $- \infty$ from the left and $\infty$ from the right, the limit does not exist.

$Does not exist$