Let $f$ be the function defined by $f(x)=2x+3e^{-5x},$ and let $g$ be a differentiable function with derivative given by $g^{\text{'}}(x)=\frac{1}{x}+4cos(\frac{5}{x}).$ It is known that $\underset{x}{lim}g(x)=.$ The value of $\underset{x}{lim}\frac{f(x)}{g(x)}$ is

$($A$)$

$\frac{1}{2}$

$1$

nonexistent