Which entire function $f\left(x\right)$ goes asymptotically to $\dfrac{e^{x}}{x}$ as $x$ goes to infinity with $x$ positive? That is, $\left(e^{x}/x \right)/f \left(x \right) \rightarrow 1$.

As mentioned in the comments, the asymptotic behavior of $f$ along the real axis doesn't really tell you anything about the function globally. For example, your function $f$ can behave in any way you like as $x\to\infty$. Indeed, the following is implied by Arakeljan's (or Nersesjan's) approximation theorem: Let $A$ be any finite union of disjoint curves tending to infinity, and let $g:A\to\mathbb{C}$ and $\varepsilon:A\to(0,\infty)$ be continuous. Then there is an entire function $f$ such that $f(z)g(z)<\varepsilon(z)$ for every $z\in A$. So, for example, let $A=(\infty,0] \cup [1,\infty)$, let $g$ be the function $e^{x}/x$ on $[1,\infty)$ and let $g$ be arbitrary on $(\infty,0]$. Then you can find an entire function that approximates this function arbitrarily closely. 

