Calculus Examples

$lim_{x \to - 8} \frac{6}{e^{x} - 3}$

Step 1

Evaluate the limit.

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Step 1.1

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

$6 lim_{x \to - 8} \frac{1}{e^{x} - 3}$

Step 1.2

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

$6 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} e^{x} - 3}$

Step 1.3

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

$6 \frac{1}{lim_{x \to - 8} e^{x} - 3}$

Step 1.4

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

$6 \frac{1}{lim_{x \to - 8} e^{x} - lim_{x \to - 8} 3}$

Step 1.5

Move the limit into the exponent.

$6 \frac{1}{e^{lim_{x \to - 8} x} - lim_{x \to - 8} 3}$

Step 1.6

Evaluate the limit of $3$ which is constant as $x$ approaches $- 8$ .

$6 \frac{1}{e^{lim_{x \to - 8} x} - 1 \cdot 3}$

Step 2

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

$6 \frac{1}{e^{- 8} - 1 \cdot 3}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

$6 \frac{1}{\frac{1}{e^{8}} - 1 \cdot 3}$

Step 3.1.2

Multiply $- 1$ by $3$ .

$6 \frac{1}{\frac{1}{e^{8}} - 3}$

Step 3.1.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{e^{8}}{e^{8}}$ .

$6 \frac{1}{\frac{1}{e^{8}} - 3 \cdot \frac{e^{8}}{e^{8}}}$

Step 3.1.4

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

$6 \frac{1}{\frac{1}{e^{8}} + \frac{- 3 e^{8}}{e^{8}}}$

Step 3.1.5

Combine the numerators over the common denominator.

$6 \frac{1}{\frac{1 - 3 e^{8}}{e^{8}}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$6 (1 \frac{e^{8}}{1 - 3 e^{8}})$

Step 3.3

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

$6 \frac{e^{8}}{1 - 3 e^{8}}$

Step 3.4

Combine $6$ and $\frac{e^{8}}{1 - 3 e^{8}}$ .

$\frac{6 e^{8}}{1 - 3 e^{8}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{6 e^{8}}{1 - 3 e^{8}}$

Decimal Form:

$- 2.00022366 \dots$