Calculus Examples

$lim_{x \to 8} \frac{5 e^{- x} + 10}{6 e^{x} - 1}$

Step 1

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

$\frac{lim_{x \to 8} 5 e^{- x} + 10}{lim_{x \to 8} 6 e^{x} - 1}$

Step 2

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

$\frac{lim_{x \to 8} 5 e^{- x} + lim_{x \to 8} 10}{lim_{x \to 8} 6 e^{x} - 1}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 8} e^{- x} + lim_{x \to 8} 10}{lim_{x \to 8} 6 e^{x} - 1}$

Step 4

Move the limit into the exponent.

$\frac{5 e^{lim_{x \to 8} - x} + lim_{x \to 8} 10}{lim_{x \to 8} 6 e^{x} - 1}$

Step 5

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

$\frac{5 e^{- lim_{x \to 8} x} + lim_{x \to 8} 10}{lim_{x \to 8} 6 e^{x} - 1}$

Step 6

Evaluate the limit of $10$ which is constant as $x$ approaches $8$ .

$\frac{5 e^{- lim_{x \to 8} x} + 10}{lim_{x \to 8} 6 e^{x} - 1}$

Step 7

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

$\frac{5 e^{- lim_{x \to 8} x} + 10}{lim_{x \to 8} 6 e^{x} - lim_{x \to 8} 1}$

Step 8

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

$\frac{5 e^{- lim_{x \to 8} x} + 10}{6 lim_{x \to 8} e^{x} - lim_{x \to 8} 1}$

Step 9

Move the limit into the exponent.

$\frac{5 e^{- lim_{x \to 8} x} + 10}{6 e^{lim_{x \to 8} x} - lim_{x \to 8} 1}$

Step 10

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

$\frac{5 e^{- lim_{x \to 8} x} + 10}{6 e^{lim_{x \to 8} x} - 1 \cdot 1}$

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 $x$ .

$\frac{5 e^{- 8} + 10}{6 e^{lim_{x \to 8} x} - 1 \cdot 1}$

Step 11.2

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

$\frac{5 e^{- 8} + 10}{6 e^{8} - 1 \cdot 1}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

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

$\frac{5 \frac{1}{e^{8}} + 10}{6 e^{8} - 1 \cdot 1}$

Step 12.1.2

Combine $5$ and $\frac{1}{e^{8}}$ .

$\frac{\frac{5}{e^{8}} + 10}{6 e^{8} - 1 \cdot 1}$

Step 12.1.3

To write $10$ as a fraction with a common denominator, multiply by $\frac{e^{8}}{e^{8}}$ .

$\frac{\frac{5}{e^{8}} + \frac{10 e^{8}}{e^{8}}}{6 e^{8} - 1 \cdot 1}$

Step 12.1.4

Combine the numerators over the common denominator.

$\frac{\frac{5 + 10 e^{8}}{e^{8}}}{6 e^{8} - 1 \cdot 1}$

Step 12.2

Multiply $- 1$ by $1$ .

$\frac{\frac{5 + 10 e^{8}}{e^{8}}}{6 e^{8} - 1}$

Step 12.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 + 10 e^{8}}{e^{8}} \cdot \frac{1}{6 e^{8} - 1}$

Step 12.4

Multiply $\frac{5 + 10 e^{8}}{e^{8}}$ by $\frac{1}{6 e^{8} - 1}$ .

$\frac{5 + 10 e^{8}}{e^{8} (6 e^{8} - 1)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{5 + 10 e^{8}}{e^{8} (6 e^{8} - 1)}$

Decimal Form:

$0.00055922 \dots$