[Math] Show limit does not exist $\lim_{z \to 0}e^{\frac{1}{z}}$

Question

How would you formally show that $\lim_{z \to 0}e^{\frac{1}{z}}$ does not exist?

I think you just show that the limit along two different paths towards the origin are different such as along $z=x$ and $z=iy$ but I don't know how to show it.

Could someone please help me out?

Answer 1

if the limit exists then any sequence tending to zero must give the same answer in the limit.

So consider $x_n = 1/n$ and $y_n = -1/n.$ One gives infinity, the other gives zero.

done.