Proving the limit of a multivariable function

Hi there! I'm newly getting introduced to multivariable functions and their limits. I had an assignment of 6 or 7 of them, and I got the majority except for the last. I've tried a few paths of approach and have concluded that the limit is indeed equal to 0. I just cannot prove it.

If , **FIXED AFTER EDIT.**

then

What would be a good choice for so that ?

But now that I look at it, it seems like I've done the process a tedious way as I'm not employing the IF statement (simply divided it out).

Any advice? Thanks in advance.

Re: Proving the limit of a multivariable function

You want , so:

.

So choose .

- Hollywood

Re: Proving the limit of a multivariable function

Hi Hollywood,

Thanks for replying, By the way, I fixed the IF statement in my OP because of a small mistake.

Secondly, I'm not sure how this choice of helps. In fact, I'm not sure if any choice of it will help. As I mentioned, I think my process to end up at just is unwise since that means the becomes unused in the proof.

Plus, the is actually in the proof. Taking the reciprocal of the would only give a and not a part. And we don't want to bound from below with , we want to do so from above.

Maybe I should restart the process all together? Any advice on how to approach this?

Re: Proving the limit of a multivariable function

To show that the limit is zero:

Given , set .

Then whenever , we have , so

And as a result,

.

I think that's a sound argument.

The in the denominator always pushes the function closer to zero, so that's why it's not a significant part of the proof.

- Hollywood

Re: Proving the limit of a multivariable function

That works perfectly Hollywood! Thanks a bunch for taking the time to help me and for explaining the process (so that I learn for similar problems in the future).

If someone can mark this as solved, it would be great. Again, thank you!