Calculus Examples

$lim_{x \to 0^{+}} (\sqrt{x}) e^{\cos (\frac{π}{x})}$

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 $0$ .

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

Move the limit into the exponent.

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

Step 2

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

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

Step 3

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

$\begin{array}{c} x & \cos (\frac{π}{x}) \\ 0.1 & 1 \\ 0.01 & 1 \\ 0.001 & - 0.66711772 \\ 0.0001 & - 0.10775036 \\ 0.00001 & 0.87550456 \\ 0.000001 & 0.98459857 \\ 0.0000001 & - 0.99950431 \\ 0.00000001 & 0.67066438 \\ 0 & 1 \\ 0 & 0.99999999 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $\cos (\frac{π}{x})$ as $x$ approaches $0$ from the right is $1$ .

$\sqrt{0} \cdot e^{1}$

Step 5

Simplify the answer.

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Step 5.1

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

$\sqrt{0^{2}} \cdot e^{1}$

Step 5.2

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

$0 \cdot e^{1}$

Step 5.3

Simplify.

$0 \cdot e$

Step 5.4

Multiply $0$ by $e$ .

$0$