Calculus Examples

$lim_{x \to 8} \frac{3 e^{- x}}{\ln (1 + 7 e^{- x})}$

Step 1

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

$3 lim_{x \to 8} \frac{e^{- x}}{\ln (1 + 7 e^{- x})}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$3 \frac{lim_{x \to 8} e^{- x}}{lim_{x \to 8} \ln (1 + 7 e^{- x})}$

Step 3

Move the limit into the exponent.

$3 \frac{e^{lim_{x \to 8} - x}}{lim_{x \to 8} \ln (1 + 7 e^{- x})}$

Step 4

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

$3 \frac{e^{- lim_{x \to 8} x}}{lim_{x \to 8} \ln (1 + 7 e^{- x})}$

Step 5

Move the limit inside the logarithm.

$3 \frac{e^{- lim_{x \to 8} x}}{\ln (lim_{x \to 8} 1 + 7 e^{- x})}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$3 \frac{e^{- lim_{x \to 8} x}}{\ln (lim_{x \to 8} 1 + lim_{x \to 8} 7 e^{- x})}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$3 \frac{e^{- lim_{x \to 8} x}}{\ln (1 + lim_{x \to 8} 7 e^{- x})}$

Step 8

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{e^{- lim_{x \to 8} x}}{\ln (1 + 7 lim_{x \to 8} e^{- x})}$

Step 9

Move the limit into the exponent.

$3 \frac{e^{- lim_{x \to 8} x}}{\ln (1 + 7 e^{lim_{x \to 8} - x})}$

Step 10

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

$3 \frac{e^{- lim_{x \to 8} x}}{\ln (1 + 7 e^{- lim_{x \to 8} x})}$

Step 11

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3 \frac{e^{- 8}}{\ln (1 + 7 e^{- lim_{x \to 8} x})}$

Step 11.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3 \frac{e^{- 8}}{\ln (1 + 7 e^{- 8})}$

Step 12

Simplify the answer.

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Step 12.1

Move $e^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 \frac{1}{\ln (1 + 7 e^{- 8}) e^{8}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 \frac{1}{\ln (1 + 7 \frac{1}{e^{8}}) e^{8}}$

Step 12.2.1.2

Combine $7$ and $\frac{1}{e^{8}}$ .

$3 \frac{1}{\ln (1 + \frac{7}{e^{8}}) e^{8}}$

Step 12.2.2

Write $1$ as a fraction with a common denominator.

$3 \frac{1}{\ln (\frac{e^{8}}{e^{8}} + \frac{7}{e^{8}}) e^{8}}$

Step 12.2.3

Combine the numerators over the common denominator.

$3 \frac{1}{\ln (\frac{e^{8} + 7}{e^{8}}) e^{8}}$

Step 12.2.4

Move $e^{8}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3 \frac{1}{\ln ((e^{8} + 7) e^{- 8}) e^{8}}$

Step 12.2.5

Rewrite $\ln ((e^{8} + 7) e^{- 8})$ as $\ln (e^{8} + 7) + \ln (e^{- 8})$ .

$3 \frac{1}{(\ln (e^{8} + 7) + \ln (e^{- 8})) e^{8}}$

Step 12.2.6

Use logarithm rules to move $- 8$ out of the exponent.

$3 \frac{1}{(\ln (e^{8} + 7) - 8 \ln (e)) e^{8}}$

Step 12.2.7

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

$3 \frac{1}{(\ln (e^{8} + 7) - 8 \cdot 1) e^{8}}$

Step 12.2.8

Multiply $- 8$ by $1$ .

$3 \frac{1}{(\ln (e^{8} + 7) - 8) e^{8}}$

Step 12.3

Combine $3$ and $\frac{1}{(\ln (e^{8} + 7) - 8) e^{8}}$ .

$\frac{3}{(\ln (e^{8} + 7) - 8) e^{8}}$

Step 12.4

Reorder factors in $\frac{3}{(\ln (e^{8} + 7) - 8) e^{8}}$ .

$\frac{3}{e^{8} (\ln (e^{8} + 7) - 8)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{e^{8} (\ln (e^{8} + 7) - 8)}$

Decimal Form:

$0.42907442 \dots$