1. ## L Hopital

Hi,

For this problem, lim x->1 x^(1/(x-1))
the answer is e, but i got the answer -1. could someone show me the steps. thanks.

Oh yes, this problem also,
lim x-> infinity xtan(1/x)

so far, I did this: lim x-> infinity x(-1/x^2 sec^2(1/x)) + tan(1/x)(1)
= lim x-> infinity -1/x sec^2(1/x) + tan(1/x)
all i did was take derivative, but not sure what to do next or maybe i started off wrong...

2. You could try transforming into something more familiar.

$\displaystyle \lim_{x\rightarrow{1}}{x^{\frac{1}{x-1}}}$

Let $\displaystyle t=\frac{1}{x-1}$

$\displaystyle x=\frac{1}{t}+1$

$\displaystyle \lim_{t\rightarrow{\infty}}\left(1+\frac{1}{t}\rig ht)^{t}$

This is a famous limit which converges to e

3. Originally Posted by ch2kb0x
Oh yes, this problem also,
lim x-> infinity xtan(1/x)
Applying L'Hopitals blindly will put you in a soup... You will never stop differentiating >_<
Instead do a little manipulation first:
Let $\displaystyle x \rightarrow \frac1{u}$
$\displaystyle \lim_{x\to \infty} x\tan\left(\frac1{x}\right) = \lim_{u\to 0} \frac{\tan(u)}{u}$