Calculus Examples

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

Step 1

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

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

Step 2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

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

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

Step 4

Move the limit into the exponent.

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $6$ which is constant as $x$ approaches $8$ .

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

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 $x$ .

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

Step 8.2

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

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

Step 9

Simplify each term.

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Step 9.1

Multiply $6$ by $8$ .

$48 \cdot e^{\frac{1}{8}} - 1 \cdot 6$

Step 9.2

Multiply $- 1$ by $6$ .

$48 e^{\frac{1}{8}} - 6$

Step 10

The result can be shown in multiple forms.

Exact Form:

$48 e^{\frac{1}{8}} - 6$

Decimal Form:

$48.39112574 \dots$