Evaluate the following limits analytically, indicating when, if necessary, L$\text{'}$H$$pital$\text{'}$s rule is used. If the limit does not exist because it is infinite, indicate which type of infinity, i$.$e$.,+$ or $-.$

a$.(3$ points$)$

$\underset{x1}{lim}\frac{sin(x)}{ln(x)}$

b$.(3$ points$)$

$\underset{x}{lim}x{e}^{-x}$

c$.(4$ points$)$ Evaluate

$\underset{x0}{lim}\frac{x}{tan(x)}$