Limit of the function $exp$ when $z→-∞$.

Question

Let us define a function $exp:C→C$ by the assignment $exp(z)=$$\sum_{n=0}^\infty (z^n/n!)$, where $z∈C$. Now it is clear that $exp$ is a holomorphic function, and hence continuous. Also it is clear from the definition of $exp$ that $exp(z)→∞ $ when $z→∞ $. Now how can I find the limit of $exp(z)$ when $z→-∞ $. The limit is seem to be zero, but how can I find(or prove) it from the definition given above. For $z→∞ $ we can easily determined its limit from the definition. But I have problem with the limit when $z→-∞$. Please help me to understand this.

Answer 1

You can show that $\exp(z+w)=\exp(z)\exp(w)$ using multiplication of series and the binomial theorem. Then $\exp(z)\exp(-z)=1$, from which you get the statements for your limits (assuming, of course, that your limits only go through real numbers).