(a) Prove that if $P(x)$ is a polynomial, then $\lim _{x ightarrow 0^{+}} P\left(\frac{1}{x} ight) e^{-1 / x}=0$. (b) Define $R_{0}(u)=1$ and, for all $ \geq 0$, define $R_{n+1} (u)=u^{2}\left(R_{n}(u)-R_{n}^{\prime} (u) ight) $. Prove that for all real numbers $x>0$ and integers $n \geq 0$, the function $R_{n}(u)$ is a polynomial in $u$ and $$ \frac{d^{n}}{d x^{n}}\left(e^{-1 / x} ight)=R_{n}\left(\frac{1}{x} ight) e^{-1 / x} . $$ (c) Define $$ g(x)=\left\{\begin{array}{11} 0,& \text { if } x \leq 0 e^{-1 / x}, & \text { if } x>0 \end{array} ight. $$ Prove that for all integers $n \geq 0$, $$ g^{(n)} (x)=\left\{\begin{array}{11} 0, & \text { if } x \leq 0, W R_{n}\left(\frac{1}{x} ight) e^{-1 / x}, & \text { if } x>0 . \end{array} ight. $$ where $R_{n}(x)$ is the function defined in part (b). (Note that the point $x=0$ requires particular attention.) (d) What is the degree- 120 Maclaurin polynomial for the function $g(x)$ defined in part (c)? CS. JG.084