Problem $2$: Find the following limits

$($a$)(5$ points$)\underset{x{0}^{+}}{lim}(1+e^x{)}^{\frac{1}{x}}$

$($b$)(5$ points$)\underset{x0}{lim}\frac{sin(x)-x}{x^3}.$

$($c$)$ points$)\underset{x-}{lim}\sqrt[\phantom{2}]{x^2+2x+3}+x.$

$($d$)$ points$)\underset{x{0}^{+}}{lim}(3^{\frac{1}{x}}+4^{\frac{1}{x}}{)}^x.$

$($e$)(5$ points$)\underset{x1}{lim}\frac{1}{lnx}-\frac{x}{x-1}.$

$($f$)(5$ points$)$ Let $a_1=-1$ and $a_2=-\frac{1}{2}$ and $a_{i+2}=a_{i+1}+a_i+0\mathrm{.}5.$ Find