Limit of a two-variable function

Homework Statement

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}}##?

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?

haruspex
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2020 Award
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
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?

Mark44
Mentor

Homework Statement

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}}##?

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

Mr Davis 97
Converting to polar form is fruitful, showing that the limit does exist.
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

haruspex
Homework Helper
Gold Member
2020 Award
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
You seem to have dropped an r.

Mr Davis 97
Ray Vickson