- #1

toforfiltum

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- 4

## Homework Statement

This is the function:

##\lim_{(x,y) \rightarrow (0,0)} \frac{(x+y)^2}{x^2+y^2}##

## Homework Equations

## The Attempt at a Solution

So for ##x \rightarrow 0## along ##y=0##, ##f(x,y)=1##

For ##y \rightarrow 0## along ##x=0##, ##f(x,y)=1## also.

But the answer says there is no limit that exists.

Is it because I didn't try approaching (0,0) using other functions?

So, if I try it for ##y=mx \rightarrow 0##,

##f(x,y) = \frac {(x+mx)^2)}{x^2+m^2x^2}##

##=\frac {(1+m)^2}{1+m^2}##, which is ##\neq 1## since ##m \neq 0##

Is this the reason why? And if so, what methods should I use to evaluate if functions truly have a limit or not? What if for another case, the above three cases gave a same answer, but there is another function, say ##y=x^2## which will show that the function actually doesn't have a limit? How do I actually know for sure if a function has a limit or not?