Calculus Examples

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

Step 1

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit into the exponent.

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

Step 5

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Multiply $8$ by $8^{2}$ by adding the exponents.

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Step 7.1.1

Multiply $8$ by $8^{2}$ .

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Step 7.1.1.1

Raise $8$ to the power of $1$ .

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

Step 7.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8^{1 + 2} \cdot e^{- 8}$

Step 7.1.2

Add $1$ and $2$ .

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

Step 7.2

Raise $8$ to the power of $3$ .

$512 \cdot e^{- 8}$

Step 7.3

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

$512 \cdot \frac{1}{e^{8}}$

Step 7.4

Combine $512$ and $\frac{1}{e^{8}}$ .

$\frac{512}{e^{8}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{512}{e^{8}}$

Decimal Form:

$0.17175686 \dots$