$(2.)$ Use l$\text{'}$H$$pital$\text{'}$s Rule to prove that the following limits exist, and evaluate the limits:

$($a$)\underset{x0}{lim}\frac{sinh(x+sinx)}{sinx}$;

$($b$)\underset{x1}{lim}\frac{(5x+3{)}^{\frac{1}{3}}-(x+3{)}^{\frac{1}{2}}}{x-1}$;

$($c$)\underset{x0}{lim}\frac{1-cosx}{x^2}$;

$($d$)\underset{x0}{lim}\frac{sinhx-x}{sin(3x^3)}.$