I'm having quite some trouble FULLY understanding this topic, and in particular this definition is difficult to get my head around:
What is the direct approach to take when trying to evaluate whether the limit exists or not?
Take this example from the textbook:
I don't understand why they've evaluated the limits by first approaching from (0,0) along the x-axis and then the y-axis if they were only going to dismiss this idea by saying "Although we have obtained identical limits along the axes, that does not show that the given limit is 0."
Why not just approach (0,0) from the x-axis and then (0,0) from y=x straight off the bat to eliminate the unecessary step of approaching (0,0) from the y-axis?
I feel like I am missing some fundamental idea here, what is it exactly?
Furthermore, what does it mean "Although we have obtained identical limits along the axes, that does not show that the given limit is 0". If the identical limits along the axes do not show this, what do they tell us then?