Does limit exists? f(x,y) = (x^2/(x^2 + (y - 2)^2)).cos(pi/y)

[ManganetaMathematica]

Hi everyone!

I'm trying to find out the limit of this function

f(x,y) = (x^2/(x^2 + (y - 2)^2)).cos(pi/y)

when (x,y) > (0,2).

Can anyone help with this?

Thanks in advance!

 

[topsquark]

Hi everyone!

I'm trying to find out the limit of this function

f(x,y) = (x^2/(x^2 + (y - 2)^2)).cos(pi/y)

when (x,y) > (0,2).

Can anyone help with this?

Thanks in advance!

You need to tell us what x and y values you want the limit to be taken. ie. What is \(\displaystyle \lim_{(x, y) \to (1, 1)} f(x, y)\)

-Dan

 

[ksdhart2]

As you (should) know, multi-variable limits can often be difficult to find because there are infinitely many "paths" you can take to approach the limit. Proving a limit doesn't exist is as simple as finding two paths that approach different limits. By contrast, proving a limit does exist must involve a systematic proof to show that every path approaches the same limit. With that in mind, a good starting point might be to just consider some possible paths. If you find two that approach different limits, you're good to go. If not, it may be time to consider other techniques, such as the multi-dimensional analogue of the squeeze theorem or a delta-epsilon proof.

So... what are your thoughts? What have you tried? Please share with us any and all work you've done on this problem, even the parts you know for sure are wrong. Thank you.

 

[mmm4444bot]

You need to tell us what x and y values you want the limit to be taken …

I'm thinking the OP used the greater-than symbol as an arrow. That is:

(x,y) > (0,2) means (x,y) --> (0,2)