Calculus Examples

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

Step 1

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

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

Step 2

Move the exponent $8$ from $x^{8}$ outside the limit using the Limits Power Rule.

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

Step 3

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

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

Step 4

Move the limit into the exponent.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

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 $x$ .

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

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

$16777216 \cdot (e^{\frac{1}{8}} - 1 \cdot 1)$

Step 9.2

Multiply $- 1$ by $1$ .

$16777216 \cdot (e^{\frac{1}{8}} - 1)$

Step 9.3

Apply the distributive property.

$16777216 e^{\frac{1}{8}} + 16777216 \cdot - 1$

Step 9.4

Multiply $16777216$ by $- 1$ .

$16777216 e^{\frac{1}{8}} - 16777216$

Step 10

The result can be shown in multiple forms.

Exact Form:

$16777216 e^{\frac{1}{8}} - 16777216$

Decimal Form:

$2233860.35716801$