Calculus Examples

$lim_{x \to 8} {(e)}^{- x} + 2 \cos (3 x)$

Step 1

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

$lim_{x \to 8} {(e)}^{- x} + lim_{x \to 8} 2 \cos (3 x)$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 8} - x} + lim_{x \to 8} 2 \cos (3 x)$

Step 3

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

$e^{- lim_{x \to 8} x} + lim_{x \to 8} 2 \cos (3 x)$

Step 4

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

$e^{- lim_{x \to 8} x} + 2 lim_{x \to 8} \cos (3 x)$

Step 5

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

$e^{- lim_{x \to 8} x} + 2 \cos (lim_{x \to 8} 3 x)$

Step 6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$e^{- lim_{x \to 8} x} + 2 \cos (3 lim_{x \to 8} x)$

Step 7

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

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Step 7.1

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

$e^{- 8} + 2 \cos (3 lim_{x \to 8} x)$

Step 7.2

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

$e^{- 8} + 2 \cos (3 \cdot 8)$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

$\frac{1}{e^{8}} + 2 \cos (3 \cdot 8)$

Step 8.1.2

Multiply $3$ by $8$ .

$\frac{1}{e^{8}} + 2 \cos (24)$

Step 8.1.3

Evaluate $\cos (24)$ .

$\frac{1}{e^{8}} + 2 \cdot 0.91354545$

Step 8.1.4

Multiply $2$ by $0.91354545$ .

$\frac{1}{e^{8}} + 1.82709091$

Step 8.2

Add $\frac{1}{e^{8}}$ and $1.82709091$ .

$1.82742637$