# 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}}##?

## 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?

## Answers and Replies

Homework Helper
Gold Member
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?

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
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

Homework Helper
Gold Member
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