Calculus Examples

$\frac{lim_{x \to 8} 11 e^{- x} + 8}{8 e^{- x} + 11}$

Step 1

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

$\frac{lim_{x \to 8} 11 e^{- x} + lim_{x \to 8} 8}{8 e^{- x} + 11}$

Step 2

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$\frac{11 lim_{x \to 8} e^{- x} + lim_{x \to 8} 8}{8 e^{- x} + 11}$

Step 3

Move the limit into the exponent.

$\frac{11 e^{lim_{x \to 8} - x} + lim_{x \to 8} 8}{8 e^{- x} + 11}$

Step 4

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

$\frac{11 e^{- lim_{x \to 8} x} + lim_{x \to 8} 8}{8 e^{- x} + 11}$

Step 5

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

$\frac{11 e^{- lim_{x \to 8} x} + 8}{8 e^{- x} + 11}$

Step 6

Simplify terms.

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Step 6.1

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

$\frac{11 e^{- 8} + 8}{8 e^{- x} + 11}$

Step 6.2

Simplify the answer.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

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

$\frac{11 \frac{1}{e^{8}} + 8}{8 e^{- x} + 11}$

Step 6.2.1.2

Combine $11$ and $\frac{1}{e^{8}}$ .

$\frac{\frac{11}{e^{8}} + 8}{8 e^{- x} + 11}$

Step 6.2.1.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{e^{8}}{e^{8}}$ .

$\frac{\frac{11}{e^{8}} + \frac{8 e^{8}}{e^{8}}}{8 e^{- x} + 11}$

Step 6.2.1.4

Combine the numerators over the common denominator.

$\frac{\frac{11 + 8 e^{8}}{e^{8}}}{8 e^{- x} + 11}$

Step 6.2.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{11 + 8 e^{8}}{e^{8}} \cdot \frac{1}{8 e^{- x} + 11}$

Step 6.2.3

Multiply $\frac{11 + 8 e^{8}}{e^{8}}$ by $\frac{1}{8 e^{- x} + 11}$ .

$\frac{11 + 8 e^{8}}{e^{8} (8 e^{- x} + 11)}$