Calculus Examples

$lim_{x \to 8} \frac{15 e^{\frac{1}{x}} - 15 x}{\frac{1}{x}}$

Step 1

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

$\frac{lim_{x \to 8} 15 e^{\frac{1}{x}} - 15 x}{lim_{x \to 8} \frac{1}{x}}$

Step 2

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

$\frac{lim_{x \to 8} 15 e^{\frac{1}{x}} - lim_{x \to 8} 15 x}{lim_{x \to 8} \frac{1}{x}}$

Step 3

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{15 lim_{x \to 8} e^{\frac{1}{x}} - lim_{x \to 8} 15 x}{lim_{x \to 8} \frac{1}{x}}$

Step 4

Move the limit into the exponent.

$\frac{15 e^{lim_{x \to 8} \frac{1}{x}} - lim_{x \to 8} 15 x}{lim_{x \to 8} \frac{1}{x}}$

Step 5

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

$\frac{15 e^{\frac{lim_{x \to 8} 1}{lim_{x \to 8} x}} - lim_{x \to 8} 15 x}{lim_{x \to 8} \frac{1}{x}}$

Step 6

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

$\frac{15 e^{\frac{1}{lim_{x \to 8} x}} - lim_{x \to 8} 15 x}{lim_{x \to 8} \frac{1}{x}}$

Step 7

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{15 e^{\frac{1}{lim_{x \to 8} x}} - 15 lim_{x \to 8} x}{lim_{x \to 8} \frac{1}{x}}$

Step 8

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

$\frac{15 e^{\frac{1}{lim_{x \to 8} x}} - 15 lim_{x \to 8} x}{\frac{lim_{x \to 8} 1}{lim_{x \to 8} x}}$

Step 9

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

$\frac{15 e^{\frac{1}{lim_{x \to 8} x}} - 15 lim_{x \to 8} x}{\frac{1}{lim_{x \to 8} x}}$

Step 10

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

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Step 10.1

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

$\frac{15 e^{\frac{1}{8}} - 15 lim_{x \to 8} x}{\frac{1}{lim_{x \to 8} x}}$

Step 10.2

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

$\frac{15 e^{\frac{1}{8}} - 15 \cdot 8}{\frac{1}{lim_{x \to 8} x}}$

Step 10.3

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

$\frac{15 e^{\frac{1}{8}} - 15 \cdot 8}{\frac{1}{8}}$

Step 11

Simplify the answer.

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Step 11.1

Multiply the numerator by the reciprocal of the denominator.

$(15 e^{\frac{1}{8}} - 15 \cdot 8) \cdot 8$

Step 11.2

Multiply $- 15$ by $8$ .

$(15 e^{\frac{1}{8}} - 120) \cdot 8$

Step 11.3

Apply the distributive property.

$15 e^{\frac{1}{8}} \cdot 8 - 120 \cdot 8$

Step 11.4

Multiply $8$ by $15$ .

$120 e^{\frac{1}{8}} - 120 \cdot 8$

Step 11.5

Multiply $- 120$ by $8$ .

$120 e^{\frac{1}{8}} - 960$

Step 12

The result can be shown in multiple forms.

Exact Form:

$120 e^{\frac{1}{8}} - 960$

Decimal Form:

$- 824.02218563 \dots$