Help finding limit (somewhat urgent): Lim (x,y)-(0,0) (x+y)^2/(x^2+y^2)

[jagijarus]

Good Afternoon

I am trying to solve a limit that by one method it says that there isnt a limit and by other it says limit is 1.

Lim (x,y)-(0,0) (x+y)^2/(x^2+y^2).

By substututing y per mx it says that the limit depends of m so there is no limit.
If i try by (x,0) then by (0,y) then both values fall on 1 so the limit is 1.

Thanks for your time
Have a great day

 

[ksdhart2]

The trouble with multivariable limits is that there are literally infinitely many paths one can take to approach the limit point. Showing that a given limit exists means finding a systematic proof to demonstrate that every path converges to the same value. Conversely, to show a limit doesn't exist can often be trivial - you need only show that two distinct paths approach different values. When you tried using the path y = mx and found that the value depends on m, you showed just that. The path y = x approaches 2, but the path y = 2x approaches 9/5. Ergo, the limit cannot exist.

On the other hand, the core flaw with your reasoning in your second method is that you've not shown a contradiction, nor have you said anything about every path. By fixing y constant and letting x approach 0, you've only shown that this one path approaches 1. Similarly, by fixing x constant and letting y approach 0, you've shown that this one path also approaches 1. Together, you've shown that these two paths both approach the same value, but that's insufficient as a proof.

 

[jagijarus]

Thank you so much, you helped me a lot