Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.06518762 \dots$