# Limit of a two-variable function

1. Mar 2, 2017

### Mr Davis 97

1. The problem statement, all variables and given/known data
Does the following limit exist: $\displaystyle \lim_{(x,y) \rightarrow (0,0)} = \frac{\sqrt{x^2+y^2+xy^2}}{\sqrt{x^2+y^2}}$?

2. Relevant equations

3. The attempt at a solution
So I am trying to evaluate the limit along several curves, such as y=x, y=0, y=x^2, and I keep getting 1. I can't think of any other curves to evaluate on. Does the limit exist or is there another curve that I am not evaluating that gives a different limit?

2. Mar 2, 2017

### haruspex

If it does exist, it does not matter how many curves you try you will not have proved it. So how about trying to prove it exists?

3. Mar 2, 2017

### Staff: Mentor

Converting to polar form is fruitful, showing that the limit does exist.

4. Mar 2, 2017

### Mr Davis 97

Converting to polar form, I get, after simplification, $\displaystyle \lim_{r \rightarrow 0^+} \sqrt{1+\cos \theta \sin^2 \theta}$. I'm not sure how to proceed

5. Mar 2, 2017

### haruspex

You seem to have dropped an r.

6. Mar 2, 2017

### Ray Vickson

Wrong expression!