Stuck on a limit calculation problem from my 20 years old Calculus 2 book.

[bundleguide]

Hi everyone !

I got the weird idea of revisiting my old math books from 20 years ago
and I got stuck on an problem that is probably super easy for most of you.

It's from the chapter where we learn about how to solve limits of the form

$$0^0, \infty^0, 1^\infty$$

by transforming an equation like this :

$$\lim_{x\rightarrow a} y = f(x)^{g(x)}$$

into this :

$$\lim_{x\rightarrow a} \ln y = g(x) \times \ln f(x)$$

etc.

The problem statement, with answer from the book :

$$\lim_{x\rightarrow0} \left[(\tan x)/x\right]^{1/x^2} = e^{1/3}$$

The attempt at a solution

It's the last problem of a serie of 50 and I got all the others right !

I really tried by myself, its true !

Anyone can help me on how to find the answer ?

Thanks !

 

[Dick]

If you got the other ones right, you're probably pretty good. Why don't you post your attempt on this one and show us why you're having problems with this one. Then we can help.

 

[ice109]

i'll give you a hint ln(x/y) = lnx - lny

 

[Gib Z]

i'll give you a hint ln(x/y) = lnx - lny

That does not help in this question. I hope (s)he is familiar with the Taylor series of the natural log and tan x, because I see no other way of doing this question.

 

[Dick]

That does not help in this question. I hope (s)he is familiar with the Taylor series of the natural log and tan x, because I see no other way of doing this question.

I think that is the best approach. l'Hopital gets pretty nasty, but I'm sure it could be done with patience. More than I have.

 

[bundleguide]

Hi again !

I don't think its really usefull to post my attempt at calculating this with Hospital's rule since the problem is that it gets very long and I hoped there was a way to simplify the formula somehow before calculating the limit or something like that.

I don't remember about Taylor series but they are explained at the end of my book.
So I will put this problem aside for now and go back to it when I know about Taylor series.

I will tell you then how things are going.

Thanks !

 

[Gib Z]

Hi again !

I don't think its really usefull to post my attempt at calculating this with Hospital's rule since the problem is that it gets very long and I hoped there was a way to simplify the formula somehow before calculating the limit or something like that.

I don't remember about Taylor series but they are explained at the end of my book.
So I will put this problem aside for now and go back to it when I know about Taylor series.

I will tell you then how things are going.

Thanks !

Excellent Idea and good luck!