$\text{ Let yey(t) be a solution to the differential equation }{\mathrm{y}}^1+2\mathrm{ty}={\mathrm{t}}^2\text{ then }16\underset{\mathrm{t}\to \infty }{\lim }\text{ y is }$

$\frac{\mathrm{dy}}{\mathrm{dt}}+\left(2\mathrm{y}\right)\mathrm{t}={\mathrm{t}}^2\to \mathrm{I}.\mathrm{F}.={\mathrm{e}}^{\int 2tdt}={\mathrm{e}}^{{\mathrm{t}}^2}$

$\begin{array}{r}\text{ Solution of the }\mathrm{D}.\mathrm{E}\text{ is }{\mathrm{ye}}^{t^2}=\mathrm{t}\frac{{\mathrm{e}}^{t^2}}{2}-\frac{1}{2}\int {\mathrm{e}}^{t^2}\mathrm{dt}+\mathrm{c}\\ \underset{t\to \infty }{\lim }\frac{\mathrm{y}}{\mathrm{t}}=\frac{1}{2}-\underset{t\to \infty }{\lim }\frac{\int {\mathrm{e}}^{{\mathrm{t}}^2}/2}{{\mathrm{te}}^{t^2}}+\frac{\mathrm{c}}{{\mathrm{te}}^{t^2}}=1/2=0.5\end{array}$