Calculus Examples

$lim_{x \to 8} e^{- 0.8 x} (1 + 5 x^{6})$

Step 1

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

$lim_{x \to 8} e^{- 0.8 x} \cdot lim_{x \to 8} 1 + 5 x^{6}$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 8} - 0.8 x} \cdot lim_{x \to 8} 1 + 5 x^{6}$

Step 3

Move the term $- 0.8$ outside of the limit because it is constant with respect to $x$ .

$e^{- 0.8 lim_{x \to 8} x} \cdot lim_{x \to 8} 1 + 5 x^{6}$

Step 4

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

$e^{- 0.8 lim_{x \to 8} x} \cdot (lim_{x \to 8} 1 + lim_{x \to 8} 5 x^{6})$

Step 5

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

$e^{- 0.8 lim_{x \to 8} x} \cdot (1 + lim_{x \to 8} 5 x^{6})$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$e^{- 0.8 lim_{x \to 8} x} \cdot (1 + 5 lim_{x \to 8} x^{6})$

Step 7

Move the exponent $6$ from $x^{6}$ outside the limit using the Limits Power Rule.

$e^{- 0.8 lim_{x \to 8} x} \cdot (1 + 5 {(lim_{x \to 8} x)}^{6})$

Step 8

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

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Step 8.1

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

$e^{- 0.8 \cdot 8} \cdot (1 + 5 {(lim_{x \to 8} x)}^{6})$

Step 8.2

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

$e^{- 0.8 \cdot 8} \cdot (1 + 5 \cdot 8^{6})$

Step 9

Simplify the answer.

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Step 9.1

Multiply $- 0.8$ by $8$ .

$e^{- 6.4} \cdot (1 + 5 \cdot 8^{6})$

Step 9.2

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

$\frac{1}{e^{6.4}} \cdot (1 + 5 \cdot 8^{6})$

Step 9.3

Simplify each term.

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Step 9.3.1

Raise $8$ to the power of $6$ .

$\frac{1}{e^{6.4}} \cdot (1 + 5 \cdot 262144)$

Step 9.3.2

Multiply $5$ by $262144$ .

$\frac{1}{e^{6.4}} \cdot (1 + 1310720)$

Step 9.4

Add $1$ and $1310720$ .

$\frac{1}{e^{6.4}} \cdot 1310721$

Step 9.5

Combine $\frac{1}{e^{6.4}}$ and $1310721$ .

$\frac{1310721}{e^{6.4}}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1310721}{e^{6.4}}$

Decimal Form:

$2177.83801065 \dots$