Answer
$$\frac{-1}{2}$$
Work Step by Step
Given $$ \lim _{t\rightarrow \infty}\frac{t-t\sqrt{t} }{2t^{3/2}+3t-5}$$
Then
\begin{aligned}
\lim _{t\rightarrow \infty}\frac{t-t\sqrt{t} }{2t^{3/2}+3t-5} &=
\lim _{t\rightarrow \infty}\frac{\frac{t}{t^{3/2}}-\frac{t\sqrt{t}}{t^{3/2}}}{\frac{2t^{3/2}}{t^{3/2}}+3\frac{t}{t^{3/2}}-\frac{5}{t^{3/2}}} \\
&= \lim _{t\rightarrow \infty}\frac{\frac{1}{t^{1/2}}-1}{2+\frac{3}{t^{1/2}}-\frac{5}{t^{3/2}}} \\
&= \lim _{t\rightarrow \infty}\frac{0-1 }{2+0-0}\\
&=\frac{-1}{2}
\end{aligned}
