Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold. Then is the following true?

$\exists C>0,\exists b\ge 2$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:

$\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} \le C\cdot t^{-b}$

What I am trying is to look at the limit of the fraction of the LHS/RHS when $t\rightarrow \infty$, and if it turns out to be finite, would that validate the conjecture?