Consider the following functions

$f(x)=sin(\frac{1}{x})\frac{1}{x-}$ and $g(x)=\{\begin{array}{ccc}\frac{e^{-x^2}}{x^2}& \text{f}\text{o}\text{r}& x<mn>0</mn>\\ x& \text{f}\text{o}\text{r}& x0\end{array}$

$($a$)$ Find all the vertical and horizontal asymptotes for both $f(x)$ and $g(x).$ Make sure you justify your answers using limits$.$

$($b$)$ Modify each function so that it retains all its vertical asymptotes but has a single horizontal asymptote at $y=1.$