Calculus Examples

$lim_{x \to 8} 7 x e^{\frac{1}{x} - 7 x}$

Step 1

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$7 lim_{x \to 8} x e^{\frac{1}{x} - 7 x}$

Step 2

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

$7 lim_{x \to 8} x \cdot lim_{x \to 8} e^{\frac{1}{x} - 7 x}$

Step 3

Move the limit into the exponent.

$7 lim_{x \to 8} x \cdot e^{lim_{x \to 8} \frac{1}{x} - 7 x}$

Step 4

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

$7 lim_{x \to 8} x \cdot e^{lim_{x \to 8} \frac{1}{x} - lim_{x \to 8} 7 x}$

Step 5

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

$7 lim_{x \to 8} x \cdot e^{\frac{lim_{x \to 8} 1}{lim_{x \to 8} x} - lim_{x \to 8} 7 x}$

Step 6

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

$7 lim_{x \to 8} x \cdot e^{\frac{1}{lim_{x \to 8} x} - lim_{x \to 8} 7 x}$

Step 7

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$7 lim_{x \to 8} x \cdot e^{\frac{1}{lim_{x \to 8} x} - 7 lim_{x \to 8} x}$

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 $x$ .

$7 \cdot 8 \cdot e^{\frac{1}{lim_{x \to 8} x} - 7 lim_{x \to 8} x}$

Step 8.2

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

$7 \cdot 8 \cdot e^{\frac{1}{8} - 7 lim_{x \to 8} x}$

Step 8.3

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

$7 \cdot 8 \cdot e^{\frac{1}{8} - 7 \cdot 8}$

Step 9

Simplify the answer.

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Step 9.1

Multiply $7$ by $8$ .

$56 \cdot e^{\frac{1}{8} - 7 \cdot 8}$

Step 9.2

Multiply $- 7$ by $8$ .

$56 \cdot e^{\frac{1}{8} - 56}$

Step 9.3

To write $- 56$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$56 \cdot e^{\frac{1}{8} - 56 \cdot \frac{8}{8}}$

Step 9.4

Combine $- 56$ and $\frac{8}{8}$ .

$56 \cdot e^{\frac{1}{8} + \frac{- 56 \cdot 8}{8}}$

Step 9.5

Combine the numerators over the common denominator.

$56 \cdot e^{\frac{1 - 56 \cdot 8}{8}}$

Step 9.6

Simplify the numerator.

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Step 9.6.1

Multiply $- 56$ by $8$ .

$56 \cdot e^{\frac{1 - 448}{8}}$

Step 9.6.2

Subtract $448$ from $1$ .

$56 \cdot e^{\frac{- 447}{8}}$

Step 9.7

Move the negative in front of the fraction.

$56 \cdot e^{- \frac{447}{8}}$

Step 9.8

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$56 \cdot \frac{1}{e^{\frac{447}{8}}}$

Step 9.9

Combine $56$ and $\frac{1}{e^{\frac{447}{8}}}$ .

$\frac{56}{e^{\frac{447}{8}}}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{56}{e^{\frac{447}{8}}}$

Decimal Form:

$3.03377837 \cdot 10^{- 23}$