Carefully solve the following series of questions. If you cannot solve an earlier part, you may still assume the result in it to solve a later part. (a) For any polynomial $p(t)$, the limit $\lim_{t \rightarrow \infty} \frac{p(t)}{e^t}$ is independent of the polynomial $p$. Justify this statement and find the value of the limit. (b) Consider the function defined by $$\begin{aligned}
q(x) &= e^{-1/x} \text{ when } x > 0 \\
&= 0 \text{ when } x = 0 \\
&= e^{1/x} \text{ when } x < 0
\end{aligned}$$ Show that $q'(0)$ exists and find its value. Why is it enough to calculate the relevant limit from only one side? (c) Now for any positive integer $n$, show that $q^{(n)}(0)$ exists and find its value. Here $q(x)$ is the function in part (b) and $q^{(n)}(0)$ denotes its $n$-th derivative at $x = 0$.
Carefully solve the following series of questions. If you cannot solve an earlier part, you may still assume the result in it to solve a later part.\\
(a) For any polynomial $p(t)$, the limit $\lim_{t \rightarrow \infty} \frac{p(t)}{e^t}$ is independent of the polynomial $p$. Justify this statement and find the value of the limit.\\
(b) Consider the function defined by
$$\begin{aligned}
q(x) &= e^{-1/x} \text{ when } x > 0 \\
&= 0 \text{ when } x = 0 \\
&= e^{1/x} \text{ when } x < 0
\end{aligned}$$
Show that $q'(0)$ exists and find its value. Why is it enough to calculate the relevant limit from only one side?\\
(c) Now for any positive integer $n$, show that $q^{(n)}(0)$ exists and find its value. Here $q(x)$ is the function in part (b) and $q^{(n)}(0)$ denotes its $n$-th derivative at $x = 0$.