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

In summary, the person is trying to solve an equation by transforming it into a different form. They got stuck on the last problem of a series of 50 and are looking for help. Anyone familiar with the Taylor series of the natural log and tan x might be able to help them.
  • #1
bundleguide
4
0
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

[tex]0^0, \infty^0, 1^\infty[/tex]

by transforming an equation like this :

[tex]\lim_{x\rightarrow a} y = f(x)^{g(x)} [/tex]

into this :

[tex]\lim_{x\rightarrow a} \ln y = g(x) \times \ln f(x) [/tex]

etc.

The problem statement, with answer from the book :

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

The attempt at a solution

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

I really tried by myself, its true !

Anyone can help me on how to find the answer ?

Thanks !
 
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  • #2
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.
 
  • #3
i'll give you a hint ln(x/y) = lnx - lny
 
  • #4
ice109 said:
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.
 
  • #5
Gib Z said:
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.
 
  • #6
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 !
 
  • #7
bundleguide said:
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!
 

1. What is a limit calculation problem in Calculus 2?

A limit calculation problem in Calculus 2 involves finding the value that a function approaches as its input approaches a specific value. This value is known as the limit and is an important concept in calculus.

2. Why am I stuck on a limit calculation problem from a Calculus 2 book?

Limit calculation problems can be challenging and require a solid understanding of the concepts and techniques used in Calculus 2. If you are stuck on a problem, it may be helpful to review your notes or seek assistance from a tutor or professor.

3. How can I approach a limit calculation problem?

There are several techniques that can be used to approach a limit calculation problem, such as direct substitution, factoring, and using L'Hopital's rule. It is important to carefully analyze the problem and choose the most appropriate technique.

4. What are some common mistakes to avoid when solving a limit calculation problem?

Some common mistakes to avoid when solving a limit calculation problem include forgetting to check for any restrictions on the function, misapplying rules or formulas, and making errors in algebraic manipulations. It is important to double-check your work and be aware of common pitfalls.

5. How can I improve my skills in solving limit calculation problems?

Practice is key when it comes to improving your skills in solving limit calculation problems. Work through a variety of problems and seek feedback on your work. Additionally, make sure to fully understand the underlying concepts and techniques used in Calculus 2.

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