Let $[t]$ denote the greatest integer $\leq t$ and $\{t\}$ denote the fractional part of $t$. The integer value of $\alpha$ for which the left hand limit of the function $f(x) = [1+x] + \frac{\alpha^{2[x]+\{x\}} + [x] - 1}{2[x]+\{x\}}$ at $x=0$ is equal to $\alpha - \frac{4}{3}$, is _________.
