$$\lim\limits_{x\to 0} \frac{(2x \tan x)}{(1-e^x)^2}$$ I have tried using l'hospital's rule to solve for 0/0 indeterminations, but with no success. Can anyone help me to find the solution to this problem? Thank you in advance for any advice.
Can't get out of a $\frac 00$ indeterminate form for this limit
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0kayte: I've edited your post (using TeX syntax). Is $x$ missing somewhere in your denominator? – 2011-11-25
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2My guess is that the denominator should be $(1 - e^{\color{Red}{x}})^2$ instead of what's written. Then you will get the $0/0$ indeterminate form as you expect. – 2011-11-25
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4I've taken the liberty of changing that $e^2$ to $e^x$. – 2011-11-25
2 Answers
Set $g(x)=2 x \tan (x)$, $h(x)=\left(1-e^x\right)^2$, $f(x)=\frac{g(x)}{h(x)}$
Then as $\lim \limits_{x \to 0}g(x)=\lim\limits_{x \to 0}h(x)=0$ you have that
$$\lim_{x \rightarrow 0}\frac{g(x)}{h(x)}=\lim_{x \rightarrow 0}\frac{g'(x)}{h'(x)}$$
if the right hand side exists by l'Hôpital. But as $\lim\limits_{x \to 0}g'(x)=\lim\limits_{x \to 0}h'(x)=0$ you have that
$$\lim\limits_{x \to 0}\frac{g'(x)}{h'(x)}=\lim\limits_{x \to 0}\frac{g''(x)}{h''(x)}$$
if the right hand side exists by l'Hôpital. But by evaluating this you can see that
$$\lim_{x \rightarrow 0}\frac{g''(x)}{h''(x)}=\frac{4}{2}=2$$
And therefore you have your limit.
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1I would have been interested in seeing how far kayte had gotten, before posting a complete solution. – 2011-11-25
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1@GerryMyerson Actually I didn't put the derivatives into my answer to let the real work be up to him. The essential "trick" is to apply L'Hôpital twice, but in my opinion you cannot really tell him that without giving my answer. – 2011-11-25
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2In my opinion, you can let OP tell you what he (or she - I've been assuming kayte is a woman's name) has done on the problem, where he got stuck. Then you're in a better position to give encouragement and maybe give the one little missing piece that unlocks the puzzle. I don't mean for this to be taken as criticism of what you have done, it's just meant to suggest another possible way of doing things. – 2011-11-25
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0Ok, next time I will not be so hasty to give full solutions :) – 2011-11-25
$${2x\tan x \over(e^x-1)^2}={2\over \cos x}\ {\sin x \over x}\Bigl({e^x -1\over x}\Bigr)^{-2}\ \to\ 2\cdot 1\cdot 1^{-2}=2\qquad (x\to0)\ .$$