Calculus Examples

$lim_{x \to 0} \sqrt{| x |} e^{\sin (\frac{π}{x})}$

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

Move the limit inside the absolute value signs.

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

Step 1.4

Move the limit into the exponent.

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

Step 2

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

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

Step 3

Consider the left sided limit.

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

Step 4

Make a table to show the behavior of the function $\sin (\frac{π}{x})$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \sin (\frac{π}{x}) \\ - 0.1 & 0 \\ - 0.01 & 0 \\ - 0.001 & - 0.85104601 \\ - 0.0001 & 0.17871591 \\ - 0.00001 & 0.97661742 \\ - 0.000001 & 0.99374486 \\ - 0.0000001 & - 0.96737158 \\ - 0.00000001 & 0.43267965 \\ 0 & 0.00000098 \\ 0 & 0.00001749 \end{array}$

Step 5

As the $x$ values approach $0$ , the function values approach $0$ . Thus, the limit of $\sin (\frac{π}{x})$ as $x$ approaches $0$ from the left is $0$ .

$0$

Step 6

Consider the right sided limit.

$lim_{x \to 0^{+}} \sin (\frac{π}{x})$

Step 7

Make a table to show the behavior of the function $\sin (\frac{π}{x})$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \sin (\frac{π}{x}) \\ 0.1 & 0 \\ 0.01 & 0 \\ 0.001 & - 0.7449523 \\ 0.0001 & 0.99417798 \\ 0.00001 & 0.48320985 \\ 0.000001 & 0.17483031 \\ 0.0000001 & - 0.03148204 \\ 0.00000001 & - 0.74176093 \\ 0 & - 0.00000098 \\ 0 & - 0.00001749 \end{array}$

Step 8

As the $x$ values approach $0$ , the function values approach $0$ . Thus, the limit of $\sin (\frac{π}{x})$ as $x$ approaches $0$ from the right is $0$ .

$\sqrt{| 0 |} \cdot e^{0}$

Step 9

Simplify the answer.

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Step 9.1

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$\sqrt{0} \cdot e^{0}$

Step 9.2

Rewrite $0$ as $0^{2}$ .

$\sqrt{0^{2}} \cdot e^{0}$

Step 9.3

Pull terms out from under the radical, assuming positive real numbers.

$0 \cdot e^{0}$

Step 9.4

Anything raised to $0$ is $1$ .

$0 \cdot 1$

Step 9.5

Multiply $0$ by $1$ .

$0$