Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

Move the limit into the exponent.

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

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 9

Move the limit into the exponent.

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

Step 10

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

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

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 $x$ .

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

Step 11.2

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

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

Step 11.3

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

$3 \frac{e^{2 \cdot 8}}{2 e^{2 \cdot 8} - e^{3 \cdot 8}}$

Step 12

Simplify the answer.

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Step 12.1

Multiply $2$ by $8$ .

$3 \frac{e^{16}}{2 e^{2 \cdot 8} - e^{3 \cdot 8}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply $2$ by $8$ .

$3 \frac{e^{16}}{2 e^{16} - e^{3 \cdot 8}}$

Step 12.2.2

Multiply $3$ by $8$ .

$3 \frac{e^{16}}{2 e^{16} - e^{24}}$

Step 12.3

Combine $3$ and $\frac{e^{16}}{2 e^{16} - e^{24}}$ .

$\frac{3 e^{16}}{2 e^{16} - e^{24}}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{3 e^{16}}{2 e^{16} - e^{24}}$

Decimal Form:

$- 0.00100706 \dots$