What is the limit as X - 0 of x^tan(x)?

Question

What is the limit as X -> 0 of x^tan(x)?

ikram002p · Answer

tan 0 =0

ikram002p · Answer

so what u have ill be 0^0
which is indeterminate .

anonymous · Answer

I think  you would do something like... $$
x^{\tan x} = e^{\tan x \ln x}
$$

anonymous · Answer

Well, $\tan x $ is undefined.

anonymous · Answer

$\tan 0$ I mean

anonymous · Answer

Or is that $\cot 0$?  Hold on...

ikram002p · Answer

ermm its zeron ;_;

anonymous · Answer

It should be a number as it involves L'Hopital's Rule but I can't seem to get past taking the natural log of both sides.

anonymous · Answer

Well, $x = e^{\ln x}$ works for $x>0$, so it can only help us for the one sided limit.

aum · Answer

$$
y = x^{tan(x)} \
\ln(y) = tan(x) * \ln(x) \
\ln(y) = \frac{\ln(x)}{\cot(x)} \
\lim_{x \rightarrow 0 } \frac{\ln(x)}{\cot(x)} = \lim_{x \rightarrow 0 }1/x / (-\cot(x)\csc(x)) = -\lim_{x \rightarrow 0 }\frac{\sin^2(x)}{x*\cos(x)} = \
-\lim_{x \rightarrow 0 } \frac{\sin(x)}{x} * \lim_{x \rightarrow 0 } \tan(x) = -1 * 0 = 0\
y = e^0 = 1
$$