JEE Main 2022 · 27 Jun · Shift 2, Mathematics question 25

Let $[t]$ denote the greatest integer $\le t$ and $\{t\}$ denote the fractional part of $t$. The 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 $\alpha-\dfrac43$, is ______.