Limit of a two-variable function

[Mr Davis 97]

Homework Statement

Does the following limit exist: $\displaystyle \lim_{(x,y) \rightarrow (0,0)} = \frac{\sqrt{x^2+y^2+xy^2}}{\sqrt{x^2+y^2}}$?

Homework Equations

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]

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]

Homework Statement

Does the following limit exist: $\displaystyle \lim_{(x,y) \rightarrow (0,0)} = \frac{\sqrt{x^2+y^2+xy^2}}{\sqrt{x^2+y^2}}$?

Homework Equations

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]

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.

 

[Ray Vickson]

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

Wrong expression!