Let [t] denote the greatest integer ≤ t and \ t\ denote the...

Given,

$f\left(x\right) = \left[1 + x\right] + \dfrac{{\alpha}^{2\left[x\right] + \left\{x\right\}} + \left[x\right] - 1}{2\left[x\right] + \left\{x\right\}}$

$\lim_{x \to {0}^{-}}f\left(x\right) = \alpha - \dfrac{4}{3}$

$\lim_{x \to {0}^{-}}\left[1 + x\right] + \dfrac{{\alpha}^{\left[x\right] + \left[x\right] + \left\{x\right\}} + \left[x\right] - 1}{2\left[x\right] + \left\{x\right\}} = \alpha - \dfrac{4}{3}$

$⇒ 0 + \dfrac{{\alpha}^{- 1} - 2}{- 1} = \alpha - \dfrac{4}{3}$

$⇒ 2 - \dfrac{1}{\alpha} = \alpha - \dfrac{4}{3}$

$⇒ \alpha + \dfrac{1}{\alpha} = \dfrac{10}{3}$

On solving above equation we get,

$⇒ \alpha = 3; \alpha \in I$