Calculus Examples

$lim_{x \to 8} 8 x (e^{\frac{9}{x}} - 1)$

Step 1

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

$8 lim_{x \to 8} x (e^{\frac{9}{x}} - 1)$

Step 2

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

$8 lim_{x \to 8} x \cdot lim_{x \to 8} e^{\frac{9}{x}} - 1$

Step 3

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

$8 lim_{x \to 8} x \cdot (lim_{x \to 8} e^{\frac{9}{x}} - lim_{x \to 8} 1)$

Step 4

Move the limit into the exponent.

$8 lim_{x \to 8} x \cdot (e^{lim_{x \to 8} \frac{9}{x}} - lim_{x \to 8} 1)$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$8 lim_{x \to 8} x \cdot (e^{9 lim_{x \to 8} \frac{1}{x}} - lim_{x \to 8} 1)$

Step 6

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

$8 lim_{x \to 8} x \cdot (e^{9 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x}} - lim_{x \to 8} 1)$

Step 7

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

$8 lim_{x \to 8} x \cdot (e^{9 \frac{1}{lim_{x \to 8} x}} - lim_{x \to 8} 1)$

Step 8

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

$8 lim_{x \to 8} x \cdot (e^{9 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 1)$

Step 9

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

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Step 9.1

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

$8 \cdot 8 \cdot (e^{9 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 1)$

Step 9.2

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

$8 \cdot 8 \cdot (e^{9 (\frac{1}{8})} - 1 \cdot 1)$

Step 10

Simplify the answer.

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Step 10.1

Multiply $8$ by $8$ .

$64 \cdot (e^{9 (\frac{1}{8})} - 1 \cdot 1)$

Step 10.2

Simplify each term.

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Step 10.2.1

Combine $9$ and $\frac{1}{8}$ .

$64 \cdot (e^{\frac{9}{8}} - 1 \cdot 1)$

Step 10.2.2

Multiply $- 1$ by $1$ .

$64 \cdot (e^{\frac{9}{8}} - 1)$

Step 10.3

Apply the distributive property.

$64 e^{\frac{9}{8}} + 64 \cdot - 1$

Step 10.4

Multiply $64$ by $- 1$ .

$64 e^{\frac{9}{8}} - 64$

Step 11

The result can be shown in multiple forms.

Exact Form:

$64 e^{\frac{9}{8}} - 64$

Decimal Form:

$133.13387833 \dots$