Calculus Examples

$lim_{x \to 8} \frac{50}{1 + 3 e^{- x}}$

Step 1

Move the term $50$ outside of the limit because it is constant with respect to $x$ .

$50 lim_{x \to 8} \frac{1}{1 + 3 e^{- x}}$

Step 2

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

$50 \frac{lim_{x \to 8} 1}{lim_{x \to 8} 1 + 3 e^{- x}}$

Step 3

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

$50 \frac{1}{lim_{x \to 8} 1 + 3 e^{- x}}$

Step 4

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

$50 \frac{1}{lim_{x \to 8} 1 + lim_{x \to 8} 3 e^{- x}}$

Step 5

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

$50 \frac{1}{1 + lim_{x \to 8} 3 e^{- x}}$

Step 6

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

$50 \frac{1}{1 + 3 lim_{x \to 8} e^{- x}}$

Step 7

Move the limit into the exponent.

$50 \frac{1}{1 + 3 e^{lim_{x \to 8} - x}}$

Step 8

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

$50 \frac{1}{1 + 3 e^{- lim_{x \to 8} x}}$

Step 9

Simplify terms.

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Step 9.1

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

$50 \frac{1}{1 + 3 e^{- 8}}$

Step 9.2

Simplify the answer.

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Step 9.2.1

Simplify the denominator.

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Step 9.2.1.1

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

$50 \frac{1}{1 + 3 \frac{1}{e^{8}}}$

Step 9.2.1.2

Combine $3$ and $\frac{1}{e^{8}}$ .

$50 \frac{1}{1 + \frac{3}{e^{8}}}$

Step 9.2.1.3

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

$50 \frac{1}{\frac{e^{8}}{e^{8}} + \frac{3}{e^{8}}}$

Step 9.2.1.4

Combine the numerators over the common denominator.

$50 \frac{1}{\frac{e^{8} + 3}{e^{8}}}$

Step 9.2.2

Multiply the numerator by the reciprocal of the denominator.

$50 (1 \frac{e^{8}}{e^{8} + 3})$

Step 9.2.3

Multiply $\frac{e^{8}}{e^{8} + 3}$ by $1$ .

$50 \frac{e^{8}}{e^{8} + 3}$

Step 9.2.4

Combine $50$ and $\frac{e^{8}}{e^{8} + 3}$ .

$\frac{50 e^{8}}{e^{8} + 3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{50 e^{8}}{e^{8} + 3}$

Decimal Form:

$49.94973119 \dots$