Calculus Examples

$lim_{x \to 8} (x^{2} - 2 x - 1) e^{x}$

Step 1

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

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

Step 2

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

$(lim_{x \to 8} x^{2} - lim_{x \to 8} 2 x - lim_{x \to 8} 1) \cdot lim_{x \to 8} e^{x}$

Step 3

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

$({(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 2 x - lim_{x \to 8} 1) \cdot lim_{x \to 8} e^{x}$

Step 4

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

$({(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x - lim_{x \to 8} 1) \cdot lim_{x \to 8} e^{x}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$({(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x - 1 \cdot 1) \cdot lim_{x \to 8} e^{x}$

Step 6

Move the limit into the exponent.

$({(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x - 1 \cdot 1) \cdot e^{lim_{x \to 8} x}$

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 $x$ .

$(8^{2} - 2 lim_{x \to 8} x - 1 \cdot 1) \cdot e^{lim_{x \to 8} x}$

Step 7.2

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

$(8^{2} - 2 \cdot 8 - 1 \cdot 1) \cdot e^{lim_{x \to 8} x}$

Step 7.3

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

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

Step 8.1.2

Multiply $- 2$ by $8$ .

$(64 - 16 - 1 \cdot 1) \cdot e^{8}$

Step 8.1.3

Multiply $- 1$ by $1$ .

$(64 - 16 - 1) \cdot e^{8}$

Step 8.2

Subtract $16$ from $64$ .

$(48 - 1) \cdot e^{8}$

Step 8.3

Subtract $1$ from $48$ .

$47 e^{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$47 e^{8}$

Decimal Form:

$140105.02539096 \dots$