In an answer [here](https://mathoverflow.net/questions/190837/entire-function-bounded-at-every-line), several users reference entire functions which are bounded along many or every line. I have two questions tending in the opposite direction:


1) Is there an entire function (not a polynomial) which diverges along every path $\gamma:[0,1)\to\mathbb{C}$ such that $|\gamma(t)|\to\infty$ as $t\to1^-$?


2) Is there an entire function $f$ which satisfies the above conclusion in the right half plane (ie. diverges along any path approaching $\infty$ which is contained in the right half plane) but converges (say to $0$) along any path approaching $\infty$ which is contained in the left half plane?



Entire function which diverges along every path
For (1), just take $f(z) = z$. (Or do you want to call that "converging to $\infty$"?)


For (2), the problem is what happens on the imaginary axis. Whether $f(it)$ diverges or converges to $0$ as $t \to +\infty$, you can take a continuous function $g: [0,\infty) \to (0,\infty)$ that goes to $0$ so rapidly that $f(it)$, $f(g(t) + it)$ and $f(-g(t) + it)$ have the same behaviour as $t \to +\infty$.
Thus you can't have one diverging and the other converging to $0$.


EDIT: For the revised question, an example is $f(z) = z \sin(z)$. 
Note that on the boundary of the rectangle with corners $\pm (n+1/2) \pi\pm n i$ for positive integers $n$, $|f(z)| \ge n$. Since $\gamma(t)$ crosses infinitely many of those boundaries, $f(\gamma(t))$ diverges.


