Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Move the limit into the exponent.

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - e^{8}}{1 + e^{x}}$

Step 3.2

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

$\frac{1^{2} - {(e^{4})}^{2}}{1 + e^{x}}$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e^{4}$ .

$\frac{(1 + e^{4}) (1 - e^{4})}{1 + e^{x}}$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{(1 + e^{4}) (1^{2} - e^{4})}{1 + e^{x}}$

Step 3.4.2

Rewrite $e^{4}$ as ${(e^{2})}^{2}$ .

$\frac{(1 + e^{4}) (1^{2} - {(e^{2})}^{2})}{1 + e^{x}}$

Step 3.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e^{2}$ .

$\frac{(1 + e^{4}) ((1 + e^{2}) (1 - e^{2}))}{1 + e^{x}}$

Step 3.4.4

Simplify.

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Step 3.4.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{(1 + e^{4}) ((1 + e^{2}) (1^{2} - e^{2}))}{1 + e^{x}}$

Step 3.4.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e$ .

$\frac{(1 + e^{4}) ((1 + e^{2}) (1 + e) (1 - e))}{1 + e^{x}}$

$\frac{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{1 + e^{x}}$