Calculus Examples

$lim_{x \to 8} (\frac{x}{6}) \cdot (e^{\frac{2}{x}} - 1)$

Step 1

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

$lim_{x \to 8} \frac{x}{6} \cdot lim_{x \to 8} e^{\frac{2}{x}} - 1$

Step 2

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{6} lim_{x \to 8} x \cdot lim_{x \to 8} e^{\frac{2}{x}} - 1$

Step 3

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

$\frac{1}{6} lim_{x \to 8} x \cdot (lim_{x \to 8} e^{\frac{2}{x}} - lim_{x \to 8} 1)$

Step 4

Move the limit into the exponent.

$\frac{1}{6} lim_{x \to 8} x \cdot (e^{lim_{x \to 8} \frac{2}{x}} - lim_{x \to 8} 1)$

Step 5

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

$\frac{1}{6} lim_{x \to 8} x \cdot (e^{2 lim_{x \to 8} \frac{1}{x}} - lim_{x \to 8} 1)$

Step 6

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

$\frac{1}{6} lim_{x \to 8} x \cdot (e^{2 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x}} - lim_{x \to 8} 1)$

Step 7

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

$\frac{1}{6} lim_{x \to 8} x \cdot (e^{2 \frac{1}{lim_{x \to 8} x}} - lim_{x \to 8} 1)$

Step 8

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

$\frac{1}{6} lim_{x \to 8} x \cdot (e^{2 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 1)$

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 $x$ .

$\frac{1}{6} \cdot 8 \cdot (e^{2 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 1)$

Step 9.2

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

$\frac{1}{6} \cdot 8 \cdot (e^{2 (\frac{1}{8})} - 1 \cdot 1)$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $2$ .

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Step 10.1.1

Factor $2$ out of $6$ .

$\frac{1}{2 (3)} \cdot 8 \cdot (e^{2 (\frac{1}{8})} - 1 \cdot 1)$

Step 10.1.2

Factor $2$ out of $8$ .

$\frac{1}{2 \cdot 3} \cdot (2 \cdot 4) \cdot (e^{2 (\frac{1}{8})} - 1 \cdot 1)$

Step 10.1.3

Cancel the common factor.

$\frac{1}{2 \cdot 3} \cdot (2 \cdot 4) \cdot (e^{2 (\frac{1}{8})} - 1 \cdot 1)$

Step 10.1.4

Rewrite the expression.

$\frac{1}{3} \cdot 4 \cdot (e^{2 (\frac{1}{8})} - 1 \cdot 1)$

Step 10.2

Combine $\frac{1}{3}$ and $4$ .

$\frac{4}{3} \cdot (e^{2 (\frac{1}{8})} - 1 \cdot 1)$

Step 10.3

Simplify each term.

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Step 10.3.1

Cancel the common factor of $2$ .

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Step 10.3.1.1

Factor $2$ out of $8$ .

$\frac{4}{3} \cdot (e^{2 \frac{1}{2 (4)}} - 1 \cdot 1)$

Step 10.3.1.2

Cancel the common factor.

$\frac{4}{3} \cdot (e^{2 \frac{1}{2 \cdot 4}} - 1 \cdot 1)$

Step 10.3.1.3

Rewrite the expression.

$\frac{4}{3} \cdot (e^{\frac{1}{4}} - 1 \cdot 1)$

Step 10.3.2

Multiply $- 1$ by $1$ .

$\frac{4}{3} \cdot (e^{\frac{1}{4}} - 1)$

Step 10.4

Apply the distributive property.

$\frac{4}{3} e^{\frac{1}{4}} + \frac{4}{3} \cdot - 1$

Step 10.5

Combine $\frac{4}{3}$ and $e^{\frac{1}{4}}$ .

$\frac{4 e^{\frac{1}{4}}}{3} + \frac{4}{3} \cdot - 1$

Step 10.6

Multiply $\frac{4}{3} \cdot - 1$ .

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Step 10.6.1

Combine $\frac{4}{3}$ and $- 1$ .

$\frac{4 e^{\frac{1}{4}}}{3} + \frac{4 \cdot - 1}{3}$

Step 10.6.2

Multiply $4$ by $- 1$ .

$\frac{4 e^{\frac{1}{4}}}{3} + \frac{- 4}{3}$

Step 10.7

Move the negative in front of the fraction.

$\frac{4 e^{\frac{1}{4}}}{3} - \frac{4}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{4 e^{\frac{1}{4}}}{3} - \frac{4}{3}$

Decimal Form:

$0.37870055 \dots$