I hope this is in the right place, it feels like calculus, but it's the last part of my analysis problem.
Construct an example where g: R2->R lim x->a g(x) exists but lim ||x||->||a|| g(x) does not exist
I'm having a very hard time coming up with something to put this together. I think this is my theory behind it, does anyone have any ideas on something that would work?
Specify a continuous functiong(x ⃗ )= g(x,y) on R^2, which is not constant, and which cannot be strictly written as a function of r = √(x^2+y^2 ). Then, the limit of g(x ⃗ ) as x ⃗ approaches a ⃗,a constant vector,will exist (and will equal g(a ⃗ ) ), but the limit of g(x ⃗ ) as |x ⃗ |approaches |a ⃗ | (a constant positive number) will not exist because x ⃗ can approach many different values in R2 (and still have|x ⃗ |approach |a ⃗ |), but the values that g(x ⃗ ) approach will be different.