Calculus Examples

$lim_{x \to 2} (x \sin (2 x)) + 10 e^{- 0.356 x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$lim_{x \to 2} x \sin (2 x) + lim_{x \to 2} 10 e^{- 0.356 x}$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $2$ .

$lim_{x \to 2} x \cdot lim_{x \to 2} \sin (2 x) + lim_{x \to 2} 10 e^{- 0.356 x}$

Step 3

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

$lim_{x \to 2} x \cdot \sin (lim_{x \to 2} 2 x) + lim_{x \to 2} 10 e^{- 0.356 x}$

Step 4

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

$lim_{x \to 2} x \cdot \sin (2 lim_{x \to 2} x) + lim_{x \to 2} 10 e^{- 0.356 x}$

Step 5

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to 2} x \cdot \sin (2 lim_{x \to 2} x) + 10 lim_{x \to 2} e^{- 0.356 x}$

Step 6

Move the limit into the exponent.

$lim_{x \to 2} x \cdot \sin (2 lim_{x \to 2} x) + 10 e^{lim_{x \to 2} - 0.356 x}$

Step 7

Move the term $- 0.356$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to 2} x \cdot \sin (2 lim_{x \to 2} x) + 10 e^{- 0.356 lim_{x \to 2} x}$

Step 8

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$2 \cdot \sin (2 lim_{x \to 2} x) + 10 e^{- 0.356 lim_{x \to 2} x}$

Step 8.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$2 \cdot \sin (2 \cdot 2) + 10 e^{- 0.356 lim_{x \to 2} x}$

Step 8.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$2 \cdot \sin (2 \cdot 2) + 10 e^{- 0.356 \cdot 2}$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $2$ by $2$ .

$2 \cdot \sin (4) + 10 e^{- 0.356 \cdot 2}$

Step 9.1.2

Evaluate $\sin (4)$ .

$2 \cdot 0.06975647 + 10 e^{- 0.356 \cdot 2}$

Step 9.1.3

Multiply $2$ by $0.06975647$ .

$0.13951294 + 10 e^{- 0.356 \cdot 2}$

Step 9.1.4

Multiply $- 0.356$ by $2$ .

$0.13951294 + 10 e^{- 0.712}$

Step 9.1.5

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

$0.13951294 + 10 \frac{1}{e^{0.712}}$

Step 9.1.6

Combine $10$ and $\frac{1}{e^{0.712}}$ .

$0.13951294 + \frac{10}{e^{0.712}}$

Step 9.2

Add $0.13951294$ and $\frac{10}{e^{0.712}}$ .

$5.04613186$