Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Move the limit inside the logarithm.

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1}$

Step 1.1.3.1.3

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

One to any power is one.

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

Step 1.1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 1} \frac{\ln (x)}{x^{2} - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.2

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

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

Step 1.3.3

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.3.5

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

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

Step 1.3.6

Add $2 x$ and $0$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Combine factors.

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Step 1.5.1

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

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

Step 1.5.2

Raise $x$ to the power of $1$ .

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

Step 1.5.3

Raise $x$ to the power of $1$ .

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

Step 1.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.5.5

Add $1$ and $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Combine.

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

Step 4.2

Multiply $1$ by $1$ .

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

Step 4.3

One to any power is one.

$\frac{1}{2 \cdot 1}$

Step 4.4

Multiply $2$ by $1$ .

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$