Calculus Examples

$lim_{x \to - 8} \frac{e^{x} - 2 x}{4 x}$

Step 1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to - 8} \frac{e^{x} - 2 x}{x}$

Step 2

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

$\frac{1}{4} \cdot \frac{lim_{x \to - 8} e^{x} - 2 x}{lim_{x \to - 8} x}$

Step 3

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

$\frac{1}{4} \cdot \frac{lim_{x \to - 8} e^{x} - lim_{x \to - 8} 2 x}{lim_{x \to - 8} x}$

Step 4

Move the limit into the exponent.

$\frac{1}{4} \cdot \frac{e^{lim_{x \to - 8} x} - lim_{x \to - 8} 2 x}{lim_{x \to - 8} x}$

Step 5

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

$\frac{1}{4} \cdot \frac{e^{lim_{x \to - 8} x} - 2 lim_{x \to - 8} x}{lim_{x \to - 8} x}$

Step 6

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

Tap for more steps...

Step 6.1

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

$\frac{1}{4} \cdot \frac{e^{- 8} - 2 lim_{x \to - 8} x}{lim_{x \to - 8} x}$

Step 6.2

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

$\frac{1}{4} \cdot \frac{e^{- 8} - 2 \cdot - 8}{lim_{x \to - 8} x}$

Step 6.3

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

$\frac{1}{4} \cdot \frac{e^{- 8} - 2 \cdot - 8}{- 8}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

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

$\frac{1}{4} \cdot \frac{\frac{1}{e^{8}} - 2 \cdot - 8}{- 8}$

Step 7.1.2

Multiply $- 2$ by $- 8$ .

$\frac{1}{4} \cdot \frac{\frac{1}{e^{8}} + 16}{- 8}$

Step 7.1.3

To write $16$ as a fraction with a common denominator, multiply by $\frac{e^{8}}{e^{8}}$ .

$\frac{1}{4} \cdot \frac{\frac{1}{e^{8}} + \frac{16 e^{8}}{e^{8}}}{- 8}$

Step 7.1.4

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{\frac{1 + 16 e^{8}}{e^{8}}}{- 8}$

Step 7.2

Combine.

$\frac{1 \frac{1 + 16 e^{8}}{e^{8}}}{4 \cdot - 8}$

Step 7.3

Multiply $\frac{1 + 16 e^{8}}{e^{8}}$ by $1$ .

$\frac{\frac{1 + 16 e^{8}}{e^{8}}}{4 \cdot - 8}$

Step 7.4

Multiply $4$ by $- 8$ .

$\frac{\frac{1 + 16 e^{8}}{e^{8}}}{- 32}$

Step 7.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{1 + 16 e^{8}}{e^{8}} \cdot \frac{1}{- 32}$

Step 7.6

Move the negative in front of the fraction.

$\frac{1 + 16 e^{8}}{e^{8}} (- \frac{1}{32})$

Step 7.7

Multiply $\frac{1 + 16 e^{8}}{e^{8}}$ by $\frac{1}{32}$ .

$- \frac{1 + 16 e^{8}}{e^{8} \cdot 32}$

Step 7.8

Move $32$ to the left of $e^{8}$ .

$- \frac{1 + 16 e^{8}}{32 e^{8}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1 + 16 e^{8}}{32 e^{8}}$

Decimal Form:

$- 0.50001048 \dots$