Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

Move the limit into the exponent.

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

Step 5

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

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

Step 6

Move the limit into the exponent.

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

Step 7

Move the limit into the exponent.

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

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 $x$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

Simplify the answer.

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Step 9.1

Subtract $e^{8}$ from $e^{8}$ .

$\frac{0}{e^{8} + e^{8}}$

Step 9.2

Add $e^{8}$ and $e^{8}$ .

$\frac{0}{2 e^{8}}$

Step 9.3

Cancel the common factor of $0$ and $2$ .

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Step 9.3.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 e^{8}}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $2 e^{8}$ .

$\frac{2 (0)}{2 (e^{8})}$

Step 9.3.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 e^{8}}$

Step 9.3.2.3

Rewrite the expression.

$\frac{0}{e^{8}}$

Step 9.4

Divide $0$ by $e^{8}$ .

$0$