Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit into the exponent.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Move the limit into the exponent.

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

Step 9

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

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

$\frac{4 - e^{8}}{4 + 3 e^{8}}$

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Rewrite $4$ as $2^{2}$ .

$\frac{2^{2} - e^{8}}{4 + 3 e^{8}}$

Step 10.2

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

$\frac{2^{2} - {(e^{4})}^{2}}{4 + 3 e^{8}}$

Step 10.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = e^{4}$ .

$\frac{(2 + e^{4}) (2 - e^{4})}{4 + 3 e^{8}}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{(2 + e^{4}) (2 - e^{4})}{4 + 3 e^{8}}$

Decimal Form:

$- 0.33273722 \dots$