Calculus Examples

$lim_{x \to 8} \frac{e^{x}}{4 + 5 e^{3 x}}$

Step 1

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

$\frac{lim_{x \to 8} e^{x}}{lim_{x \to 8} 4 + 5 e^{3 x}}$

Step 2

Move the limit into the exponent.

$\frac{e^{lim_{x \to 8} x}}{lim_{x \to 8} 4 + 5 e^{3 x}}$

Step 3

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

$\frac{e^{lim_{x \to 8} x}}{lim_{x \to 8} 4 + lim_{x \to 8} 5 e^{3 x}}$

Step 4

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{e^{lim_{x \to 8} x}}{4 + lim_{x \to 8} 5 e^{3 x}}$

Step 5

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

$\frac{e^{lim_{x \to 8} x}}{4 + 5 lim_{x \to 8} e^{3 x}}$

Step 6

Move the limit into the exponent.

$\frac{e^{lim_{x \to 8} x}}{4 + 5 e^{lim_{x \to 8} 3 x}}$

Step 7

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

$\frac{e^{lim_{x \to 8} x}}{4 + 5 e^{3 lim_{x \to 8} x}}$

Step 8

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 8.1

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

$\frac{e^{8}}{4 + 5 e^{3 lim_{x \to 8} x}}$

Step 8.2

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

$\frac{e^{8}}{4 + 5 e^{3 \cdot 8}}$

Step 9

Multiply $3$ by $8$ .

$\frac{e^{8}}{4 + 5 e^{24}}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{8}}{4 + 5 e^{24}}$

Decimal Form:

$2.25070349 \cdot 10^{- 8}$