Calculus Examples

$lim_{n \to 8} \frac{e^{n} + e^{- n}}{e^{2 n} - 2}$

Step 1

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

$\frac{lim_{n \to 8} e^{n} + e^{- n}}{lim_{n \to 8} e^{2 n} - 2}$

Step 2

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

$\frac{lim_{n \to 8} e^{n} + lim_{n \to 8} e^{- n}}{lim_{n \to 8} e^{2 n} - 2}$

Step 3

Move the limit into the exponent.

$\frac{e^{lim_{n \to 8} n} + lim_{n \to 8} e^{- n}}{lim_{n \to 8} e^{2 n} - 2}$

Step 4

Move the limit into the exponent.

$\frac{e^{lim_{n \to 8} n} + e^{lim_{n \to 8} - n}}{lim_{n \to 8} e^{2 n} - 2}$

Step 5

Move the term $- 1$ outside of the limit because it is constant with respect to $n$ .

$\frac{e^{lim_{n \to 8} n} + e^{- lim_{n \to 8} n}}{lim_{n \to 8} e^{2 n} - 2}$

Step 6

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

$\frac{e^{lim_{n \to 8} n} + e^{- lim_{n \to 8} n}}{lim_{n \to 8} e^{2 n} - lim_{n \to 8} 2}$

Step 7

Move the limit into the exponent.

$\frac{e^{lim_{n \to 8} n} + e^{- lim_{n \to 8} n}}{e^{lim_{n \to 8} 2 n} - lim_{n \to 8} 2}$

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $n$ .

$\frac{e^{lim_{n \to 8} n} + e^{- lim_{n \to 8} n}}{e^{2 lim_{n \to 8} n} - lim_{n \to 8} 2}$

Step 9

Evaluate the limit of $2$ which is constant as $n$ approaches $8$ .

$\frac{e^{lim_{n \to 8} n} + e^{- lim_{n \to 8} n}}{e^{2 lim_{n \to 8} n} - 1 \cdot 2}$

Step 10

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

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Step 10.1

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

$\frac{e^{8} + e^{- lim_{n \to 8} n}}{e^{2 lim_{n \to 8} n} - 1 \cdot 2}$

Step 10.2

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

$\frac{e^{8} + e^{- 8}}{e^{2 lim_{n \to 8} n} - 1 \cdot 2}$

Step 10.3

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

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

Step 11.1.2

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

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

Step 11.1.3

Combine the numerators over the common denominator.

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

Step 11.1.4

Multiply $e^{8}$ by $e^{8}$ by adding the exponents.

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Step 11.1.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11.1.4.2

Add $8$ and $8$ .

$\frac{\frac{e^{16} + 1}{e^{8}}}{e^{2 \cdot 8} - 1 \cdot 2}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $2$ by $8$ .

$\frac{\frac{e^{16} + 1}{e^{8}}}{e^{16} - 1 \cdot 2}$

Step 11.2.2

Multiply $- 1$ by $2$ .

$\frac{\frac{e^{16} + 1}{e^{8}}}{e^{16} - 2}$

Step 11.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{e^{16} + 1}{e^{8}} \cdot \frac{1}{e^{16} - 2}$

Step 11.4

Multiply $\frac{e^{16} + 1}{e^{8}}$ by $\frac{1}{e^{16} - 2}$ .

$\frac{e^{16} + 1}{e^{8} (e^{16} - 2)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{16} + 1}{e^{8} (e^{16} - 2)}$

Decimal Form:

$0.00033546 \dots$