Let $[t]$ denote the greatest integer $\leq t$ and $\{t\}$ denote the fractional part of $t$. Then integral 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 $\_\_\_\_$
