Calculus Examples

$lim_{x \to 8} \frac{3 e^{2 x}}{- e^{3 x}}$

Step 1

Cancel the common factor of $e^{2 x}$ and $e^{3 x}$ .

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Step 1.1

Factor $e^{3 x}$ out of $3 e^{2 x}$ .

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

Step 1.2

Move the negative one from the denominator of $\frac{3 e^{- x}}{- 1}$ .

$lim_{x \to 8} - 1 \cdot (3 e^{- x})$

Step 2

Move the term $- 1 \cdot 3$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

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

$- 3 e^{- 8}$

Step 5.2

Simplify the answer.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.00100638 \dots$