Calculus Examples

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

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.

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

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 term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.2.1.2

Move the limit inside the logarithm.

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

Step 1.2.1.3

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

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

Step 1.2.1.4

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

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

Step 1.2.1.5

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $3$ by $1$ .

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

Step 1.2.3.1.2

Multiply $- 1$ by $2$ .

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

Step 1.2.3.2

Subtract $2$ from $3$ .

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

Step 1.2.3.3

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

$\frac{4 \cdot 0}{lim_{x \to 1} 2 - 2 x^{2}}$

Step 1.2.3.4

Multiply $4$ by $0$ .

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

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 $1$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.1.4

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

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

Step 1.3.2

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

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

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.

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

Step 1.3.3.1.2

Multiply $- 2$ by $1$ .

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

Step 1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (3 x - 2)$ with respect to $x$ is $4 \frac{d}{d x} [\ln (3 x - 2)]$ .

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

Step 3.3

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 3.3.1

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

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

Step 3.3.2

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

$lim_{x \to 1} \frac{4 (\frac{1}{u} \frac{d}{d x} [3 x - 2])}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.3.3

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

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

Step 3.4

Combine $\frac{1}{3 x - 2}$ and $4$ .

$lim_{x \to 1} \frac{\frac{4}{3 x - 2} \frac{d}{d x} [3 x - 2]}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.5

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

$lim_{x \to 1} \frac{\frac{4}{3 x - 2} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2])}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.6

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

$lim_{x \to 1} \frac{\frac{4}{3 x - 2} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.7

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

$lim_{x \to 1} \frac{\frac{4}{3 x - 2} (3 \cdot 1 + \frac{d}{d x} [- 2])}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.8

Multiply $3$ by $1$ .

$lim_{x \to 1} \frac{\frac{4}{3 x - 2} (3 + \frac{d}{d x} [- 2])}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.9

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

$lim_{x \to 1} \frac{\frac{4}{3 x - 2} (3 + 0)}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.10

Add $3$ and $0$ .

$lim_{x \to 1} \frac{\frac{4}{3 x - 2} \cdot 3}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.11

Combine $\frac{4}{3 x - 2}$ and $3$ .

$lim_{x \to 1} \frac{\frac{4 \cdot 3}{3 x - 2}}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.12

Multiply $4$ by $3$ .

$lim_{x \to 1} \frac{\frac{12}{3 x - 2}}{\frac{d}{d x} [2 - 2 x^{2}]}$

Step 3.13

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

$lim_{x \to 1} \frac{\frac{12}{3 x - 2}}{\frac{d}{d x} [2] + \frac{d}{d x} [- 2 x^{2}]}$

Step 3.14

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

$lim_{x \to 1} \frac{\frac{12}{3 x - 2}}{0 + \frac{d}{d x} [- 2 x^{2}]}$

Step 3.15

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

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Step 3.15.1

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

$lim_{x \to 1} \frac{\frac{12}{3 x - 2}}{0 - 2 \frac{d}{d x} [x^{2}]}$

Step 3.15.2

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

$lim_{x \to 1} \frac{\frac{12}{3 x - 2}}{0 - 2 (2 x)}$

Step 3.15.3

Multiply $2$ by $- 2$ .

$lim_{x \to 1} \frac{\frac{12}{3 x - 2}}{0 - 4 x}$

Step 3.16

Subtract $4 x$ from $0$ .

$lim_{x \to 1} \frac{\frac{12}{3 x - 2}}{- 4 x}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 1} \frac{12}{3 x - 2} \cdot \frac{1}{- 4 x}$

Step 5

Simplify terms.

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Step 5.1

Multiply $\frac{12}{3 x - 2}$ by $\frac{1}{- 4 x}$ .

$lim_{x \to 1} \frac{12}{(3 x - 2) (- 4 x)}$

Step 5.2

Cancel the common factor of $12$ and $- 4$ .

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Step 5.2.1

Factor $4$ out of $12$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 15

Simplify the answer.

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Step 15.1

Cancel the common factor of $1$ .

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Step 15.1.1

Cancel the common factor.

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

Step 15.1.2

Rewrite the expression.

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

Step 15.2

Cancel the common factor of $1$ and $- 1$ .

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Step 15.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 15.2.2

Move the negative in front of the fraction.

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

Step 15.3

Simplify the denominator.

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Step 15.3.1

Multiply $- 1$ by $2$ .

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

Step 15.3.2

Subtract $2$ from $3$ .

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

Step 15.4

Cancel the common factor of $1$ .

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Step 15.4.1

Cancel the common factor.

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

Step 15.4.2

Rewrite the expression.

$3 (- 1 \cdot 1)$

Step 15.5

Multiply $3 (- 1 \cdot 1)$ .

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Step 15.5.1

Multiply $- 1$ by $1$ .

$3 \cdot - 1$

Step 15.5.2

Multiply $3$ by $- 1$ .

$- 3$