Answer
$$\lim\limits_{t\to\infty}\frac{t-t\sqrt t}{2t^{3/2}+3t-5}=\frac{-1}{2}$$
Work Step by Step
$$A=\lim\limits_{t\to\infty}\frac{t-t\sqrt t}{2t^{3/2}+3t-5}$$$$A=\lim\limits_{t\to\infty}\frac{t-t\sqrt t}{(2\times t\times t^{1/2})+3t-5}$$$$A=\lim\limits_{t\to\infty}\frac{t-t\sqrt t}{2t\sqrt t+3t-5}$$
Divide both numerator and denominator by $t\sqrt t$
We also have $t\sqrt t=t\times t^{1/2}=t^{3/2}$$$A=\lim\limits_{t\to\infty}\frac{\frac{t}{t\sqrt t}-\frac{t\sqrt t}{t\sqrt t}}{\frac{2t\sqrt t}{t\sqrt t}+\frac{3t}{t\sqrt t}-\frac{5}{t\sqrt t}}$$$$A=\lim\limits_{t\to\infty}\frac{\frac{1}{\sqrt t}-1}{2+\frac{3}{\sqrt t}-\frac{5}{t\sqrt t}}$$$$A=\lim\limits_{t\to\infty}\frac{\frac{1}{t^{1/2}}-1}{2+\frac{3}{t^{1/2}}-\frac{5}{t^{3/2}}}$$$$A=\frac{0-1}{2+3\times0-5\times0}$$$$A=\frac{-1}{2}$$
