Solving the limit!! LIMIT x -> 0+ sqrt(x) * e^(sin(pi/x))

**kram19** · February 17th 2013, 12:10 PM

Help with SOLVING!!! {LIMIT x -> 0+} sqrt(x) * e^(sin(pi/x))

I know the Limit is zero I want to know how. Here's what I got:

{LIMIT x -> 0+} sqrt(x) * e^(sin(pi/x)) (I assume continuity so i plug n chug)

{LIMIT x -> 0+} sqrt(0) * e^(sin(pi/0)) => {LIMIT x -> 0+} 0 * e^(sin(pi/x)) = INDETERMINATE

{LIMIT x -> 0+} sqrt(x) * e^((pi/x)*(sin(pi/x))/pi/x)) ({LIMIT x -> 0+} {LIMIT x -> pi/x+} (sin(pi/x))/pi/x) = 1)

{LIMIT x -> 0+} sqrt(x) * e^(pi/x) (I'm stuck here i plug in 0 for x and i get e^ (pi/0))

**veileen** · February 17th 2013, 12:42 PM

$\lim_{x\rightarrow0+}\frac{\sin \frac{\pi}{x}}{\frac{\pi}{x}}$ is not necessarily 1, because $\frac{\pi}{x}\xrightarrow[]{x\rightarrow 0+}\infty$ .

The remarkable limit is $\lim_{x\rightarrow 0}\frac{\sin x}{x}=1$ .

Edit:

$-1\leq \sin \frac{\pi}{x}\leq 1 \Rightarrow e^{-1} \leq e^ {\sin \frac{\pi}{x}} \leq e$

$\Rightarrow e^{-1}\sqrt x \leq e^ {\sin \frac{\pi}{x}}\sqrt x \leq e\sqrt x$

$\lim_{x\rightarrow 0+} e^{-1}\sqrt x=0$
$\lim_{x\rightarrow 0+} e\sqrt x=0$

So the limit is 0.

**kram19** · February 17th 2013, 01:34 PM

Thank you for the reasonable answer.

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