Calculus Examples

$lim_{x \to 8} e^{- 4 x} \sin (4 x)$

Step 1

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

$lim_{x \to 8} e^{- 4 x} \cdot lim_{x \to 8} \sin (4 x)$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 8} - 4 x} \cdot lim_{x \to 8} \sin (4 x)$

Step 3

Move the term $- 4$ outside of the limit because it is constant with respect to $x$ .

$e^{- 4 lim_{x \to 8} x} \cdot lim_{x \to 8} \sin (4 x)$

Step 4

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

$e^{- 4 lim_{x \to 8} x} \cdot \sin (lim_{x \to 8} 4 x)$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$e^{- 4 lim_{x \to 8} x} \cdot \sin (4 lim_{x \to 8} x)$

Step 6

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

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

$e^{- 4 \cdot 8} \cdot \sin (4 lim_{x \to 8} x)$

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

$e^{- 4 \cdot 8} \cdot \sin (4 \cdot 8)$

Step 7

Simplify the answer.

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Multiply $- 4$ by $8$ .

$e^{- 32} \cdot \sin (4 \cdot 8)$

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

$\frac{1}{e^{32}} \cdot \sin (4 \cdot 8)$

Multiply $4$ by $8$ .

$\frac{1}{e^{32}} \cdot \sin (32)$

Evaluate $\sin (32)$ .

$\frac{1}{e^{32}} \cdot 0.52991926$

Combine $\frac{1}{e^{32}}$ and $0.52991926$ .

$\frac{0.52991926}{e^{32}}$

Replace $e$ with an approximation.

$\frac{0.52991926}{{2.71828182}^{32}}$

Raise $2.71828182$ to the power of $32$ .

$\frac{0.52991926}{78962960182676}$

Divide $0.52991926$ by $78962960182676$ .

$0$