Formal Proof for a Limit

Nov 2014
64
0
Kansas City, Kansas
limx->0 xtan(1/x). Formal proof and, if the limit exists, state the limit.

It makes sense to me that the limit doesn't exist since (1/x) becomes arbitrarily large as x->0 but I'm a little unsure how to go about proving this. I can start by breaking it up into two separate limits, but I'm not sure if that will help at all. That will just give me 0*infinity which is indeterminate.

I think I'll split it and look at tan(1/x) as sin(1/x)/cos(1/x).
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
limx->0 xtan(1/x). Formal proof and, if the limit exists, state the limit.

It makes sense to me that the limit doesn't exist since (1/x) becomes arbitrarily large as x->0 but I'm a little unsure how to go about proving this.
Therefore, what? The whole point of "limits" is to be able to work with things that "become arbitrarily large" or "go to infinity".

I can start by breaking it up into two separate limits, but I'm not sure if that will help at all. That will just give me 0*infinity which is indeterminate.

I think I'll split it and look at tan(1/x) as sin(1/x)/cos(1/x).
That's a good idea. You might also note that as x goes to 0, 1/x goes to infinity, so letting y= 1/x, this becomes "limit as y goes to infinity of \(\displaystyle \frac{sin(y)}{y}\frac{1}{cos(y)}\).

However, the problem asks you to give a "formal proof" so that if you do find a limit, you must then use a "\(\displaystyle \delta\), \(\displaystyle \epsilon\)" argument. And if you determine there is no limit, show why such an argument cannot work.
 
Nov 2014
64
0
Kansas City, Kansas
How should I set up the epsilon-delta argument?
 
Dec 2013
2,000
757
Colombia
First you need to know what limit you have (if any).

Never underestimate the power of graphing your function when trying to determine a limit. It is not a proof, but it does at least stop you from chasing the wrong answer for hours. Here's \(\displaystyle {1\over y}\tan{y}\) as set out by HallsofIvy:
 
  • Like
Reactions: 1 person