Answer
limit does not exist
Work Step by Step
Here, we have $f(x,y)=-\dfrac{x}{\sqrt{x^2+y^2}}$
Let us consider one approach : $(x,y) \to (0,0)$ along $y=kx$
Then, we get $\lim\limits_{(x,y) \to (0,0)}-\dfrac{x}{\sqrt{x^2+(kx)^2}}=\lim\limits_{(x,y) \to (0,0)}-\dfrac{1}{\sqrt{1+k^2}}$
This shows that there are multiple limit values, therefore, and so, the limit does not exist at the point (0,0) for the function $f(x,y)=-\dfrac{x}{\sqrt{x^2+y^2}}$.
