Calculus Examples

$lim_{x \to - 3} \frac{4 \ln (- 2 - x)}{3 e^{2 x + 6} - 3}$

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.

$\frac{lim_{x \to - 3} 4 \ln (- 2 - x)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{4 lim_{x \to - 3} \ln (- 2 - x)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.2

Move the limit inside the logarithm.

$\frac{4 \ln (lim_{x \to - 3} - 2 - x)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{4 \ln (- lim_{x \to - 3} 2 - lim_{x \to - 3} x)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.4

Evaluate the limit of $2$ which is constant as $x$ approaches $- 3$ .

$\frac{4 \ln (- 1 \cdot 2 - lim_{x \to - 3} x)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.5

Simplify terms.

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Step 1.2.5.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{4 \ln (- 1 \cdot 2 + 3)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.5.2

Simplify the answer.

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Step 1.2.5.2.1

Multiply $- 1$ by $2$ .

$\frac{4 \ln (- 2 + 3)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.5.2.2

Add $- 2$ and $3$ .

$\frac{4 \ln (1)}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.5.2.3

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

$\frac{4 \cdot 0}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

Step 1.2.5.2.4

Multiply $4$ by $0$ .

$\frac{0}{lim_{x \to - 3} 3 e^{2 x + 6} - 3}$

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 $- 3$ .

$\frac{0}{lim_{x \to - 3} 3 e^{2 x + 6} - lim_{x \to - 3} 3}$

Step 1.3.1.2

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

$\frac{0}{3 lim_{x \to - 3} e^{2 x + 6} - lim_{x \to - 3} 3}$

Step 1.3.1.3

Move the limit into the exponent.

$\frac{0}{3 e^{lim_{x \to - 3} 2 x + 6} - lim_{x \to - 3} 3}$

Step 1.3.1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{0}{3 e^{lim_{x \to - 3} 2 x + lim_{x \to - 3} 6} - lim_{x \to - 3} 3}$

Step 1.3.1.5

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

$\frac{0}{3 e^{2 lim_{x \to - 3} x + lim_{x \to - 3} 6} - lim_{x \to - 3} 3}$

Step 1.3.1.6

Evaluate the limit of $6$ which is constant as $x$ approaches $- 3$ .

$\frac{0}{3 e^{2 lim_{x \to - 3} x + 6} - lim_{x \to - 3} 3}$

Step 1.3.1.7

Evaluate the limit of $3$ which is constant as $x$ approaches $- 3$ .

$\frac{0}{3 e^{2 lim_{x \to - 3} x + 6} - 1 \cdot 3}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{0}{3 e^{2 \cdot - 3 + 6} - 1 \cdot 3}$

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

Multiply $2$ by $- 3$ .

$\frac{0}{3 e^{- 6 + 6} - 1 \cdot 3}$

Step 1.3.3.1.2

Add $- 6$ and $6$ .

$\frac{0}{3 e^{0} - 1 \cdot 3}$

Step 1.3.3.1.3

Anything raised to $0$ is $1$ .

$\frac{0}{3 \cdot 1 - 1 \cdot 3}$

Step 1.3.3.1.4

Multiply $3$ by $1$ .

$\frac{0}{3 - 1 \cdot 3}$

Step 1.3.3.1.5

Multiply $- 1$ by $3$ .

$\frac{0}{3 - 3}$

Step 1.3.3.2

Subtract $3$ from $3$ .

$\frac{0}{0}$

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.

$lim_{x \to - 3} \frac{4 \ln (- 2 - x)}{3 e^{2 x + 6} - 3} = lim_{x \to - 3} \frac{\frac{d}{d x} [4 \ln (- 2 - x)]}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to - 3} \frac{\frac{d}{d x} [4 \ln (- 2 - x)]}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.2

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

$lim_{x \to - 3} \frac{4 \frac{d}{d x} [\ln (- 2 - x)]}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

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

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Step 3.3.1

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

$lim_{x \to - 3} \frac{4 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [- 2 - x])}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.3.2

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

$lim_{x \to - 3} \frac{4 (\frac{1}{u} \frac{d}{d x} [- 2 - x])}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.3.3

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

$lim_{x \to - 3} \frac{4 (\frac{1}{- 2 - x} \frac{d}{d x} [- 2 - x])}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.4

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

$lim_{x \to - 3} \frac{\frac{4}{- 2 - x} \frac{d}{d x} [- 2 - x]}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.5

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

$lim_{x \to - 3} \frac{\frac{4}{- 2 - x} (\frac{d}{d x} [- 2] + \frac{d}{d x} [- x])}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.6

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

$lim_{x \to - 3} \frac{\frac{4}{- 2 - x} (0 + \frac{d}{d x} [- x])}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.7

Add $0$ and $\frac{d}{d x} [- x]$ .

$lim_{x \to - 3} \frac{\frac{4}{- 2 - x} \frac{d}{d x} [- x]}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.8

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

$lim_{x \to - 3} \frac{\frac{4}{- 2 - x} (- \frac{d}{d x} [x])}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.9

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

$lim_{x \to - 3} \frac{\frac{4}{- 2 - x} (- 1 \cdot 1)}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.10

Multiply $- 1$ by $1$ .

$lim_{x \to - 3} \frac{\frac{4}{- 2 - x} \cdot - 1}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.11

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

$lim_{x \to - 3} \frac{\frac{4 \cdot - 1}{- 2 - x}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.12

Multiply $4$ by $- 1$ .

$lim_{x \to - 3} \frac{\frac{- 4}{- 2 - x}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.13

Move the negative in front of the fraction.

$lim_{x \to - 3} \frac{- \frac{4}{- 2 - x}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.14

Simplify.

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Step 3.14.1

Rewrite $- 2$ as $- 1 (2)$ .

$lim_{x \to - 3} \frac{- \frac{4}{- 1 (2) - x}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.14.2

Factor $- 1$ out of $- x$ .

$lim_{x \to - 3} \frac{- \frac{4}{- 1 (2) - (x)}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.14.3

Factor $- 1$ out of $- 1 (2) - (x)$ .

$lim_{x \to - 3} \frac{- \frac{4}{- 1 (2 + x)}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.14.4

Move the negative in front of the fraction.

$lim_{x \to - 3} \frac{- - \frac{4}{2 + x}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.14.5

Multiply $- 1$ by $- 1$ .

$lim_{x \to - 3} \frac{1 \frac{4}{2 + x}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.14.6

Multiply $\frac{4}{2 + x}$ by $1$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{\frac{d}{d x} [3 e^{2 x + 6} - 3]}$

Step 3.15

By the Sum Rule, the derivative of $3 e^{2 x + 6} - 3$ with respect to $x$ is $\frac{d}{d x} [3 e^{2 x + 6}] + \frac{d}{d x} [- 3]$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{\frac{d}{d x} [3 e^{2 x + 6}] + \frac{d}{d x} [- 3]}$

Step 3.16

Evaluate $\frac{d}{d x} [3 e^{2 x + 6}]$ .

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Step 3.16.1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{2 x + 6}$ with respect to $x$ is $3 \frac{d}{d x} [e^{2 x + 6}]$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 \frac{d}{d x} [e^{2 x + 6}] + \frac{d}{d x} [- 3]}$

Step 3.16.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x + 6$ .

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Step 3.16.2.1

To apply the Chain Rule, set $u$ as $2 x + 6$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x + 6]) + \frac{d}{d x} [- 3]}$

Step 3.16.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{u} \frac{d}{d x} [2 x + 6]) + \frac{d}{d x} [- 3]}$

Step 3.16.2.3

Replace all occurrences of $u$ with $2 x + 6$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{2 x + 6} \frac{d}{d x} [2 x + 6]) + \frac{d}{d x} [- 3]}$

Step 3.16.3

By the Sum Rule, the derivative of $2 x + 6$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [6]$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{2 x + 6} (\frac{d}{d x} [2 x] + \frac{d}{d x} [6])) + \frac{d}{d x} [- 3]}$

Step 3.16.4

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

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{2 x + 6} (2 \frac{d}{d x} [x] + \frac{d}{d x} [6])) + \frac{d}{d x} [- 3]}$

Step 3.16.5

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

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{2 x + 6} (2 \cdot 1 + \frac{d}{d x} [6])) + \frac{d}{d x} [- 3]}$

Step 3.16.6

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{2 x + 6} (2 \cdot 1 + 0)) + \frac{d}{d x} [- 3]}$

Step 3.16.7

Multiply $2$ by $1$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{2 x + 6} (2 + 0)) + \frac{d}{d x} [- 3]}$

Step 3.16.8

Add $2$ and $0$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (e^{2 x + 6} \cdot 2) + \frac{d}{d x} [- 3]}$

Step 3.16.9

Move $2$ to the left of $e^{2 x + 6}$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{3 (2 e^{2 x + 6}) + \frac{d}{d x} [- 3]}$

Step 3.16.10

Multiply $2$ by $3$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{6 e^{2 x + 6} + \frac{d}{d x} [- 3]}$

Step 3.17

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{6 e^{2 x + 6} + 0}$

Step 3.18

Add $6 e^{2 x + 6}$ and $0$ .

$lim_{x \to - 3} \frac{\frac{4}{2 + x}}{6 e^{2 x + 6}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to - 3} \frac{4}{2 + x} \cdot \frac{1}{6 e^{2 x + 6}}$

Step 5

Simplify terms.

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Step 5.1

Multiply $\frac{4}{2 + x}$ by $\frac{1}{6 e^{2 x + 6}}$ .

$lim_{x \to - 3} \frac{4}{(2 + x) (6 e^{2 x + 6})}$

Step 5.2

Cancel the common factor of $4$ and $6$ .

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Step 5.2.1

Factor $2$ out of $4$ .

$lim_{x \to - 3} \frac{2 (2)}{(2 + x) (6 e^{2 x + 6})}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $(2 + x) (6 e^{2 x + 6})$ .

$lim_{x \to - 3} \frac{2 (2)}{2 ((2 + x) (3 e^{2 x + 6}))}$

Step 5.2.2.2

Cancel the common factor.

$lim_{x \to - 3} \frac{2 \cdot 2}{2 ((2 + x) (3 e^{2 x + 6}))}$

Step 5.2.2.3

Rewrite the expression.

$lim_{x \to - 3} \frac{2}{(2 + x) (3 e^{2 x + 6})}$

Step 6

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

$\frac{2}{3} lim_{x \to - 3} \frac{1}{(2 + x) (e^{2 x + 6})}$

Step 7

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 3$ .

$\frac{2}{3} \cdot \frac{lim_{x \to - 3} 1}{lim_{x \to - 3} (2 + x) (e^{2 x + 6})}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $- 3$ .

$\frac{2}{3} \cdot \frac{1}{lim_{x \to - 3} (2 + x) (e^{2 x + 6})}$

Step 9

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{2}{3} \cdot \frac{1}{lim_{x \to - 3} 2 + x \cdot lim_{x \to - 3} e^{2 x + 6}}$

Step 10

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{2}{3} \cdot \frac{1}{(lim_{x \to - 3} 2 + lim_{x \to - 3} x) \cdot lim_{x \to - 3} e^{2 x + 6}}$

Step 11

Evaluate the limit of $2$ which is constant as $x$ approaches $- 3$ .

$\frac{2}{3} \cdot \frac{1}{(2 + lim_{x \to - 3} x) \cdot lim_{x \to - 3} e^{2 x + 6}}$

Step 12

Move the limit into the exponent.

$\frac{2}{3} \cdot \frac{1}{(2 + lim_{x \to - 3} x) \cdot e^{lim_{x \to - 3} 2 x + 6}}$

Step 13

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{2}{3} \cdot \frac{1}{(2 + lim_{x \to - 3} x) \cdot e^{lim_{x \to - 3} 2 x + lim_{x \to - 3} 6}}$

Step 14

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

$\frac{2}{3} \cdot \frac{1}{(2 + lim_{x \to - 3} x) \cdot e^{2 lim_{x \to - 3} x + lim_{x \to - 3} 6}}$

Step 15

Evaluate the limit of $6$ which is constant as $x$ approaches $- 3$ .

$\frac{2}{3} \cdot \frac{1}{(2 + lim_{x \to - 3} x) \cdot e^{2 lim_{x \to - 3} x + 6}}$

Step 16

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 16.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{2}{3} \cdot \frac{1}{(2 - 3) \cdot e^{2 lim_{x \to - 3} x + 6}}$

Step 16.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{2}{3} \cdot \frac{1}{(2 - 3) \cdot e^{2 \cdot - 3 + 6}}$

Step 17

Simplify the answer.

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Step 17.1

Combine.

$\frac{2 \cdot 1}{3 ((2 - 3) \cdot e^{2 \cdot - 3 + 6})}$

Step 17.2

Multiply $2$ by $1$ .

$\frac{2}{3 (2 - 3) \cdot e^{2 \cdot - 3 + 6}}$

Step 17.3

Simplify the denominator.

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Step 17.3.1

Multiply $2$ by $- 3$ .

$\frac{2}{3 (2 - 3) e^{- 6 + 6}}$

Step 17.3.2

Subtract $3$ from $2$ .

$\frac{2}{3 \cdot - 1 e^{- 6 + 6}}$

Step 17.3.3

Combine exponents.

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Step 17.3.3.1

Factor out negative.

$\frac{2}{- (3 e^{- 6 + 6})}$

Step 17.3.3.2

Multiply $3$ by $- 1$ .

$\frac{2}{- 3 e^{- 6 + 6}}$

Step 17.3.4

Add $- 6$ and $6$ .

$\frac{2}{- 3 e^{0}}$

Step 17.3.5

Anything raised to $0$ is $1$ .

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

Step 17.4

Multiply $- 3$ by $1$ .

$\frac{2}{- 3}$

Step 17.5

Move the negative in front of the fraction.

$- \frac{2}{3}$