Calculus Examples

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

Step 1

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

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

Step 2

Move the limit into the exponent.

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

Step 3

Move the term $- 8$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Multiply $- 8$ by $8$ .

$e^{- 64} \cdot \cos (8)$

Step 6.2

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

$\frac{1}{e^{64}} \cdot \cos (8)$

Step 6.3

Evaluate $\cos (8)$ .

$\frac{1}{e^{64}} \cdot 0.99026806$

Step 6.4

Combine $\frac{1}{e^{64}}$ and $0.99026806$ .

$\frac{0.99026806}{e^{64}}$

Step 6.5

Replace $e$ with an approximation.

$\frac{0.99026806}{{2.71828182}^{64}}$

Step 6.6

Raise $2.71828182$ to the power of $64$ .

$\frac{0.99026806}{6235149080810880000000000000}$

Step 6.7

Divide $0.99026806$ by $6235149080810880000000000000$ .

$0$