TITLE: Find $\lim\limits_{(x,y)\to (0,0)}{\frac{e^{x}+y-1}{x+y}}$
QUESTION [6 upvotes]: Find $$\lim_{(x,y)\to (0,0)}{\dfrac{e^{x}+y-1}{x+y}}$$

I tried with different trajectories and I always get that the limit is $1$, but I cannot prove it, any help?

REPLY [5 votes]: Hint: Along the path
$$
y=1-e^x
$$
we get the limit is $0$. However, along the path
$$
x=0
$$
the limit is $1$.