I have an examples book with a limit exercise that I can't understand.
The limit in question is:
$$f(x,y)=\frac{x}{x+y}$$ with $x\ne-y$; $$\lim_{(x,y)\to(0,0)} f(x,y)$$
And then to solve it, it goes:
$$\lim_{(x,y)\to(0,0)} f(x,y) = \lim_{x\to0} f(x,mx) =\lim_{x\to 0}\frac{x}{x+mx}=\frac{1}{1+m}.$$
Can you help me understand that? Thanks,
UPDATE: Ok, just to make sure that I got it right. I have a very similar test exercise with $4$ different options.
The following limit $$\lim_{(x,y)\to(0,0)}\frac{-x^3+3xy^2}{x^2+y^2}$$ equals:
A. $0$
B. $- \infty$
C. Doesn't exist
D. $ +\infty$
My doubt is: if I consider it normally I'd say that it doesn't exist, but if I solve it using the same approach (i.e. $y=mx$) then the limit equals $0$. Which one is the right answer?