Calculus Examples

$lim_{x \to 1} \frac{4 \ln (3 x - 2)}{2 - 2 x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

Cancel the common factor of $4$ and $2 - 2 x^{2}$ .

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Step 1.1.1

Factor $2$ out of $4 \ln (3 x - 2)$ .

$lim_{x \to 1} \frac{2 (2 \ln (3 x - 2))}{2 - 2 x^{2}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $2$ .

$lim_{x \to 1} \frac{2 (2 \ln (3 x - 2))}{2 (1) - 2 x^{2}}$

Step 1.1.2.2

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

$lim_{x \to 1} \frac{2 (2 \ln (3 x - 2))}{2 (1) + 2 (- x^{2})}$

Step 1.1.2.3

Factor $2$ out of $2 (1) + 2 (- x^{2})$ .

$lim_{x \to 1} \frac{2 (2 \ln (3 x - 2))}{2 (1 - x^{2})}$

Step 1.1.2.4

Cancel the common factor.

$lim_{x \to 1} \frac{2 (2 \ln (3 x - 2))}{2 (1 - x^{2})}$

Step 1.1.2.5

Rewrite the expression.

$lim_{x \to 1} \frac{2 \ln (3 x - 2)}{1 - x^{2}}$

Step 1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 1} \frac{\ln (3 x - 2)}{1 - x^{2}}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$2 \frac{lim_{x \to 1} \ln (3 x - 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Move the limit inside the logarithm.

$2 \frac{\ln (lim_{x \to 1} 3 x - 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2.1.2

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

$2 \frac{\ln (lim_{x \to 1} 3 x - lim_{x \to 1} 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2.1.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{\ln (3 lim_{x \to 1} x - lim_{x \to 1} 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2.1.4

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$2 \frac{\ln (3 lim_{x \to 1} x - 1 \cdot 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2.2

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

$2 \frac{\ln (3 \cdot 1 - 1 \cdot 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $3$ by $1$ .

$2 \frac{\ln (3 - 1 \cdot 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2.3.1.2

Multiply $- 1$ by $2$ .

$2 \frac{\ln (3 - 2)}{lim_{x \to 1} 1 - x^{2}}$

Step 2.1.2.3.2

Subtract $2$ from $3$ .

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

Step 2.1.2.3.3

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

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

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

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

Step 2.1.3.1.2

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

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

Step 2.1.3.1.3

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

One to any power is one.

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

Step 2.1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.3.3.2

Subtract $1$ from $1$ .

$2 (\frac{0}{0})$

Step 2.1.3.3.3

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

Undefined

$2 (\frac{0}{0})$

Step 2.1.3.4

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

Undefined

$2 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$2 (\frac{0}{0})$

Step 2.2

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.

$lim_{x \to 1} \frac{\ln (3 x - 2)}{1 - x^{2}} = lim_{x \to 1} \frac{\frac{d}{d x} [\ln (3 x - 2)]}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$2 lim_{x \to 1} \frac{\frac{d}{d x} [\ln (3 x - 2)]}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.2

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 $g (x) = 3 x - 2$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $3 x - 2$ .

$2 lim_{x \to 1} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x - 2]}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.2.2

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

$2 lim_{x \to 1} \frac{\frac{1}{u} \frac{d}{d x} [3 x - 2]}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.2.3

Replace all occurrences of $u$ with $3 x - 2$ .

$2 lim_{x \to 1} \frac{\frac{1}{3 x - 2} \frac{d}{d x} [3 x - 2]}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.3

By the Sum Rule, the derivative of $3 x - 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$ .

$2 lim_{x \to 1} \frac{\frac{1}{3 x - 2} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2])}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$2 lim_{x \to 1} \frac{\frac{1}{3 x - 2} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.5

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

$2 lim_{x \to 1} \frac{\frac{1}{3 x - 2} (3 \cdot 1 + \frac{d}{d x} [- 2])}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.6

Multiply $3$ by $1$ .

$2 lim_{x \to 1} \frac{\frac{1}{3 x - 2} (3 + \frac{d}{d x} [- 2])}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 lim_{x \to 1} \frac{\frac{1}{3 x - 2} (3 + 0)}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.8

Add $3$ and $0$ .

$2 lim_{x \to 1} \frac{\frac{1}{3 x - 2} \cdot 3}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.9

Combine $\frac{1}{3 x - 2}$ and $3$ .

$2 lim_{x \to 1} \frac{\frac{3}{3 x - 2}}{\frac{d}{d x} [1 - x^{2}]}$

Step 2.3.10

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

$2 lim_{x \to 1} \frac{\frac{3}{3 x - 2}}{\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]}$

Step 2.3.11

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

$2 lim_{x \to 1} \frac{\frac{3}{3 x - 2}}{0 + \frac{d}{d x} [- x^{2}]}$

Step 2.3.12

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 2.3.12.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$2 lim_{x \to 1} \frac{\frac{3}{3 x - 2}}{0 - \frac{d}{d x} [x^{2}]}$

Step 2.3.12.2

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

$2 lim_{x \to 1} \frac{\frac{3}{3 x - 2}}{0 - 1 \cdot 2 x}$

Step 2.3.12.3

Multiply $2$ by $- 1$ .

$2 lim_{x \to 1} \frac{\frac{3}{3 x - 2}}{0 - 2 x}$

Step 2.3.13

Subtract $2 x$ from $0$ .

$2 lim_{x \to 1} \frac{\frac{3}{3 x - 2}}{- 2 x}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$2 lim_{x \to 1} \frac{3}{3 x - 2} \cdot \frac{1}{- 2 x}$

Step 2.5

Multiply $\frac{3}{3 x - 2}$ by $\frac{1}{- 2 x}$ .

$2 lim_{x \to 1} \frac{3}{(3 x - 2) \cdot - 2 x}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $\frac{3}{- 2}$ outside of the limit because it is constant with respect to $x$ .

$2 (\frac{3}{- 2}) lim_{x \to 1} \frac{1}{(3 x - 2) x}$

Step 3.2

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

$2 (\frac{3}{- 2}) \frac{lim_{x \to 1} 1}{lim_{x \to 1} (3 x - 2) x}$

Step 3.3

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

$2 (\frac{3}{- 2}) \frac{1}{lim_{x \to 1} (3 x - 2) x}$

Step 3.4

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

$2 (\frac{3}{- 2}) \frac{1}{lim_{x \to 1} 3 x - 2 \cdot lim_{x \to 1} x}$

Step 3.5

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

$2 (\frac{3}{- 2}) \frac{1}{(lim_{x \to 1} 3 x - lim_{x \to 1} 2) \cdot lim_{x \to 1} x}$

Step 3.6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$2 (\frac{3}{- 2}) \frac{1}{(3 lim_{x \to 1} x - lim_{x \to 1} 2) \cdot lim_{x \to 1} x}$

Step 3.7

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$2 (\frac{3}{- 2}) \frac{1}{(3 lim_{x \to 1} x - 1 \cdot 2) \cdot lim_{x \to 1} x}$

Step 4

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

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Step 4.1

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

$2 (\frac{3}{- 2}) \frac{1}{(3 \cdot 1 - 1 \cdot 2) \cdot lim_{x \to 1} x}$

Step 4.2

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

$2 (\frac{3}{- 2}) \frac{1}{(3 \cdot 1 - 1 \cdot 2) \cdot 1}$

Step 5

Simplify the answer.

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Step 5.1

Cancel the common factor of $2$ .

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Step 5.1.1

Factor $2$ out of $- 2$ .

$2 \frac{3}{2 (- 1)} \frac{1}{(3 \cdot 1 - 1 \cdot 2) \cdot 1}$

Step 5.1.2

Cancel the common factor.

$2 \frac{3}{2 \cdot - 1} \frac{1}{(3 \cdot 1 - 1 \cdot 2) \cdot 1}$

Step 5.1.3

Rewrite the expression.

$\frac{3}{- 1} \cdot \frac{1}{(3 \cdot 1 - 1 \cdot 2) \cdot 1}$

Step 5.2

Divide $3$ by $- 1$ .

$- 3 \frac{1}{(3 \cdot 1 - 1 \cdot 2) \cdot 1}$

Step 5.3

Cancel the common factor of $1$ .

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Step 5.3.1

Cancel the common factor.

$- 3 \frac{1}{(3 \cdot 1 - 1 \cdot 2) \cdot 1}$

Step 5.3.2

Rewrite the expression.

$- 3 \frac{1}{3 \cdot 1 - 1 \cdot 2}$

Step 5.4

Simplify the denominator.

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Step 5.4.1

Multiply $3$ by $1$ .

$- 3 \frac{1}{3 - 1 \cdot 2}$

Step 5.4.2

Multiply $- 1$ by $2$ .

$- 3 \frac{1}{3 - 2}$

Step 5.4.3

Subtract $2$ from $3$ .

$- 3 (\frac{1}{1})$

Step 5.5

Cancel the common factor of $1$ .

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Step 5.5.1

Cancel the common factor.

$- 3 (\frac{1}{1})$

Step 5.5.2

Rewrite the expression.

$- 3 \cdot 1$

Step 5.6

Multiply $- 3$ by $1$ .

$- 3$