For limits of 2 variables, I understand that I need to evaluate the limit from different paths such as if I approach (0,0) from the x-axis or the y-axis or any y = mx or any other path. I know that the limit does NOT exist if I get two different values if I take the limit as I approach (0,0) from two different paths. However, I'm having trouble finding the limit if it does indeed exist. Say it's a nice function that factors out nicely which lets me plug in the values for x, that's fine. However, I don't understand how to use the Epsilon-Delta Definition to find the limit. Can anyone help me understand? How do I choose the relationship between Epsilon and Delta for it to fit nicely?
In some problems, they make an observation and use it to find the limit. These observations make sense to me mathematically, but I don't see how one realizes that specific observation to be useful.
I've checked some of the other questions on the site but I couldn't quite make sense of everything.
Sample Problem with solution I don't quite understand.
How do I know that the observation $x^2 \le x^2 + y^2 $ will help me? How does one CHOOSE what observation to make?