Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$2 e^{lim_{x \to - 8} x} - {(lim_{x \to - 8} x)}^{2} \cdot lim_{x \to - 8} e^{x}$

Step 6

Move the limit into the exponent.

$2 e^{lim_{x \to - 8} x} - {(lim_{x \to - 8} x)}^{2} \cdot e^{lim_{x \to - 8} x}$

Step 7

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

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Step 7.1

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

$2 e^{- 8} - {(lim_{x \to - 8} x)}^{2} \cdot e^{lim_{x \to - 8} x}$

Step 7.2

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

$2 e^{- 8} - {(- 8)}^{2} \cdot e^{lim_{x \to - 8} x}$

Step 7.3

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

$2 e^{- 8} - {(- 8)}^{2} \cdot e^{- 8}$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

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

Step 8.1.2

Combine $2$ and $\frac{1}{e^{8}}$ .

$\frac{2}{e^{8}} - {(- 8)}^{2} \cdot e^{- 8}$

Step 8.1.3

Raise $- 8$ to the power of $2$ .

$\frac{2}{e^{8}} - 1 \cdot 64 \cdot e^{- 8}$

Step 8.1.4

Multiply $- 1$ by $64$ .

$\frac{2}{e^{8}} - 64 \cdot e^{- 8}$

Step 8.1.5

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

$\frac{2}{e^{8}} - 64 \cdot \frac{1}{e^{8}}$

Step 8.1.6

Combine $- 64$ and $\frac{1}{e^{8}}$ .

$\frac{2}{e^{8}} + \frac{- 64}{e^{8}}$

Step 8.1.7

Move the negative in front of the fraction.

$\frac{2}{e^{8}} - \frac{64}{e^{8}}$

Step 8.2

Combine the numerators over the common denominator.

$\frac{2 - 64}{e^{8}}$

Step 8.3

Subtract $64$ from $2$ .

$\frac{- 62}{e^{8}}$

Step 8.4

Move the negative in front of the fraction.

$- \frac{62}{e^{8}}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{62}{e^{8}}$

Decimal Form:

$- 0.02079868 \dots$