Calculus Examples

$lim_{x \to 0} e^{- x} \sin (π x)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} e^{- x} \cdot lim_{x \to 0} \sin (π x)$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 0} - x} \cdot lim_{x \to 0} \sin (π x)$

Step 3

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

$e^{- lim_{x \to 0} x} \cdot lim_{x \to 0} \sin (π x)$

Step 4

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

$e^{- lim_{x \to 0} x} \cdot \sin (lim_{x \to 0} π x)$

Step 5

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

$e^{- lim_{x \to 0} x} \cdot \sin (π lim_{x \to 0} x)$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{0} \cdot \sin (π lim_{x \to 0} x)$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{0} \cdot \sin (π \cdot 0)$

Step 7

Simplify the answer.

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Step 7.1

Anything raised to $0$ is $1$ .

$1 \cdot \sin (π \cdot 0)$

Step 7.2

Multiply $\sin (π \cdot 0)$ by $1$ .

$\sin (π \cdot 0)$

Step 7.3

Multiply $π$ by $0$ .

$\sin (0)$

Step 7.4

The exact value of $\sin (0)$ is $0$ .

$0$