Calculus Examples

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

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.

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Move the limit inside the logarithm.

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

One to any power is one.

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

Step 1.2.3.2

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

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

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 $1$ .

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

Step 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 1.3.1.3

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

One to any power is one.

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

Step 1.3.3.1.2

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 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 $g (x) = x^{2}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

Combine $2$ and $\frac{1}{x^{2}}$ .

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

Step 3.5

Combine $\frac{2}{x^{2}}$ and $x$ .

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

Step 3.6

Cancel the common factor of $x$ and $x^{2}$ .

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Step 3.6.1

Factor $x$ out of $2 x$ .

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

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $x$ out of $x^{2}$ .

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

Step 3.6.2.2

Cancel the common factor.

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

Step 3.6.2.3

Rewrite the expression.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $2 x$ and $0$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine factors.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

Evaluate the limit.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

One to any power is one.

$\frac{1}{1}$

Step 8.2

Divide $1$ by $1$ .

$1$