Prove that $\lim_{t\rightarrow0^{+}}\frac{e^{-\frac{1}{t}}}{t^{k}}=0 $ for every positive integer $k$ without L'Hospital's Rule. Prove that $\lim_{t\rightarrow0^{+}}\frac{e^{-\frac{1}{t}}}{t^{k}}=0 $  for every positive integer $k$. I am trying to show that the function $f:\mathbb{R}\rightarrow\mathbb{R}$
  defined by $f(t)=\begin{cases}
\exp\left(-\frac{1}{t}\right), & \mbox{if }t>0\\
0, & \mbox{if }t\leq0
\end{cases}$ is smooth and to prove this, I have to prove that the limit exists. I am having trouble showing that it exists. I know that it can be shown using L'Hospital's Rule, but i would prefer to show it using the $\epsilon -  \delta $ definition.

Pick $\epsilon > 0$. By taking the derivative one sees that $e^{-1/t}t^{-k}$ is increasing on some interval $(0,a)$. Choose $m \in \mathbb{N}$ such that $2^{-m} < a$ and
$$mk - 2^m < \log_2\epsilon.$$
Therefore if $0 < t < 2^{-m}$ we have
$$\frac{e^{-1/t}}{t^k} < \frac{2^{-1/t}}{t^k} < \frac{2^{mk}}{2^{2^m}} = 2^{mk - 2^m} < \epsilon.$$
