What is the Limit of a Function in Two Variables at the Origin?

[smatik]

Homework Statement

lim (x,y)\rightarrow(0,0) f(x,y)=2*x/(x^{2} + x +y^{2})

Homework Equations

used different paths like y=k*x ,where k is a constant and y=k*x^2

The Attempt at a Solution

Got an answer 2 but solution says does not exist. Can anybody convince me that why limit does not exist,without using polar form.

I am currently studying in 1st year of college.

 

[haruspex]

The problem with the paths you chose is that there is one direction you are unable to represent with a finite k. What direction do you think that would be?

 

[smatik]

oh got it thanks man. its the y axis! The limit comes out to be zero if i put x=1. Thanks!