why $\lim_{M\to\infty} e^{-kM}=0$

Question

why $\lim_{M\to\infty} e^{-kM} = 0$ if $k>0$

I know that $e^{-\infty} = 0$

but why $e^{-k\infty} = 0$

I was told I could not do operations with infinity

Answer 1

The claim is only true for $k>0$, in this case you can use the fact that $x^k$ is continuous and so

$$\lim_{M\rightarrow\infty} e^{-kM} = \lim_{M\rightarrow\infty} (e^{-M})^k = \left(\lim_{M\rightarrow\infty} e^{-M}\right)^k = 0^k = 0$$