Thread: Systematic approach for multi variable limits?

1. Systematic approach for multi variable limits?

I feel like my knowledge in finding limits of multi variable functions has a few loose ends and was wondering if anyone here might help me tie them up. I have no problem performing the various tests; if for example we wish to evaluate a limit at the origin of a function f(x,y) you can let the f approach (0,0) along the axises, a line y=kx, a curve y=x^2 and so on. But that's kind of the thing. I feel like I'm never entirely sure what tests I should use or when I have enough of them agree on a specific limit to be able to conclude that that is indeed the limit. I generally resort to graphing and that works, of course, but is there a systematic way of approaching it with just the function definition, a piece of paper, a pen and a glaring lack of artistic talent?

2. Re: Systematic approach for multi variable limits?

Originally Posted by Scurmicurv
I feel like my knowledge in finding limits of multi variable functions has a few loose ends and was wondering if anyone here might help me tie them up. I have no problem performing the various tests; if for example we wish to evaluate a limit at the origin of a function f(x,y) you can let the f approach (0,0) along the axises, a line y=kx, a curve y=x^2 and so on. But that's kind of the thing. I feel like I'm never entirely sure what tests I should use or when I have enough of them agree on a specific limit to be able to conclude that that is indeed the limit. I generally resort to graphing and that works, of course, but is there a systematic way of approaching it with just the function definition, a piece of paper, a pen and a glaring lack of artistic talent?
An approach working about 'cent per cent' is the following: use polar coordinates setting...

$x= r\ \cos \theta$

$y= r\ \sin \theta$ (1)

... and then find the limit for r to 0 independently from $\theta$. If the limit exists then You are ok!...

Marry Christmas from Serbia

$\chi$ $\sigma$

3. Re: Systematic approach for multi variable limits?

Originally Posted by chisigma
An approach working about 'cent per cent' is the following: use polar coordinates setting...

$x= r\ \cos \theta$

$y= r\ \sin \theta$ (1)

... and then find the limit for r to 0 independently from $\theta$. If the limit exists then You are ok!...

Marry Christmas from Serbia

$\chi$ $\sigma$
If the limit exists and is a number, then you have the limit.

If the limit exists and depends on \displaystyle \begin{align*} \theta \end{align*}, then you know that the limit is going to change depending on which path you take, which means the limit does not exist.

4. Re: Systematic approach for multi variable limits?

Well, that makes sense. Just as a follow-up then, there is no equivalently sure-fire way of doing it in Cartesian coordinates? How come?

5. Re: Systematic approach for multi variable limits?

Originally Posted by Scurmicurv
Well, that makes sense. Just as a follow-up then, there is no equivalently sure-fire way of doing it in Cartesian coordinates? How come?
Sometimes something similar to the approach in the following example works. You have the function...

$f(x,y)=\begin{cases}\frac{x y^{2}}{x^{2}+y^{2}} &\text{if}\ (x,y) \ne (0,0)\\ 0 &\text{if}\ (x,y) = (0,0)\end{cases}$ (1)

... and You should find $\lim_{(x,y) \rightarrow (0,0)} f(x,y)$. In that case You consider that $|x y^{2}| \le |x|\ |x^{2}+y^{2}|$ and You can write...

$|\frac{x y^{2}}{x^{2}+y^{2}}| \le |x| \implies \lim_{(x,y) \rightarrow (0,0)} f(x,y) \le \lim_{(x,y) \rightarrow (0,0)} |x|= 0$ (2)

The use of polar coordinates however is also in that case easier and faster and doesn't require to search more or less complicated 'excamotages'. The pratical limit of that approach is when the function has more than two variables. For a three variables function may be that the use of r, $\theta$ and $\phi$ sferical coordinates is possible. But if the variables are four or more?...

Marry Christmas from Serbia

$\chi$ $\sigma$

6. Re: Systematic approach for multi variable limits?

Hm. Well, food for thought, if nothing else, and those loose ends I mentioned at the outset don't feel quite so random now. Thanks!