Calculus Examples

$lim_{n \to 8} e^{- n} \cos (n)$

Step 1

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

$lim_{n \to 8} e^{- n} \cdot lim_{n \to 8} \cos (n)$

Step 2

Move the limit into the exponent.

$e^{lim_{n \to 8} - n} \cdot lim_{n \to 8} \cos (n)$

Step 3

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

$e^{- lim_{n \to 8} n} \cdot lim_{n \to 8} \cos (n)$

Step 4

Move the limit inside the trig function because cosine is continuous.

$e^{- lim_{n \to 8} n} \cdot \cos (lim_{n \to 8} n)$

Step 5

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

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Step 5.1

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

$e^{- 8} \cdot \cos (lim_{n \to 8} n)$

Step 5.2

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

$e^{- 8} \cdot \cos (8)$

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Evaluate $\cos (8)$ .

$\frac{1}{e^{8}} \cdot 0.99026806$

Step 6.3

Combine $\frac{1}{e^{8}}$ and $0.99026806$ .

$\frac{0.99026806}{e^{8}}$

Step 6.4

Replace $e$ with an approximation.

$\frac{0.99026806}{{2.71828182}^{8}}$

Step 6.5

Raise $2.71828182$ to the power of $8$ .

$\frac{0.99026806}{2980.95798704}$

Step 6.6

Divide $0.99026806$ by $2980.95798704$ .

$0.00033219$