Calculus Examples

$lim_{x \to 8} \frac{6 e^{x}}{1 + e^{- x}}$

Step 1

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 lim_{x \to 8} \frac{e^{x}}{1 + e^{- x}}$

Step 2

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

$6 \frac{lim_{x \to 8} e^{x}}{lim_{x \to 8} 1 + e^{- x}}$

Step 3

Move the limit into the exponent.

$6 \frac{e^{lim_{x \to 8} x}}{lim_{x \to 8} 1 + e^{- x}}$

Step 4

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

$6 \frac{e^{lim_{x \to 8} x}}{lim_{x \to 8} 1 + lim_{x \to 8} e^{- x}}$

Step 5

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

$6 \frac{e^{lim_{x \to 8} x}}{1 + lim_{x \to 8} e^{- x}}$

Step 6

Move the limit into the exponent.

$6 \frac{e^{lim_{x \to 8} x}}{1 + e^{lim_{x \to 8} - x}}$

Step 7

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

$6 \frac{e^{lim_{x \to 8} x}}{1 + e^{- lim_{x \to 8} x}}$

Step 8

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

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Step 8.1

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

$6 \frac{e^{8}}{1 + e^{- lim_{x \to 8} x}}$

Step 8.2

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

$6 \frac{e^{8}}{1 + e^{- 8}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

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

$6 \frac{e^{8}}{1 + \frac{1}{e^{8}}}$

Step 9.1.2

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

$6 \frac{e^{8}}{\frac{e^{8}}{e^{8}} + \frac{1}{e^{8}}}$

Step 9.1.3

Combine the numerators over the common denominator.

$6 \frac{e^{8}}{\frac{e^{8} + 1}{e^{8}}}$

Step 9.2

Multiply the numerator by the reciprocal of the denominator.

$6 (e^{8} \frac{e^{8}}{e^{8} + 1})$

Step 9.3

Multiply $e^{8} \frac{e^{8}}{e^{8} + 1}$ .

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Step 9.3.1

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

$6 \frac{e^{8} e^{8}}{e^{8} + 1}$

Step 9.3.2

Multiply $e^{8}$ by $e^{8}$ by adding the exponents.

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Step 9.3.2.1

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

$6 \frac{e^{8 + 8}}{e^{8} + 1}$

Step 9.3.2.2

Add $8$ and $8$ .

$6 \frac{e^{16}}{e^{8} + 1}$

Step 9.4

Combine $6$ and $\frac{e^{16}}{e^{8} + 1}$ .

$\frac{6 e^{16}}{e^{8} + 1}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{6 e^{16}}{e^{8} + 1}$

Decimal Form:

$17879.74993435 \dots$