Mathematics - Limits Question with Solution | TestHub

$\begin{array}{rl}& \underset{t\to x}{\lim }\frac{t^{2}f(x)-x^{2}f(t)}{t-x}=1\\ & \underset{t\to x}{\lim }\frac{2t.f(x)-x^2{f}^{\prime }(x)}{1}=1\\ & 2x\cdot f(x)-x2f^{\prime }(x)=1\\ & \frac{dy}{dx}-\frac{2}{x}\cdot y=\frac{-1}{x^2}\\ & \text{ I.f. }=e^{\int -\frac{2}{x}dx}=\frac{1}{x^2}\\ & \therefore \frac{y}{x^2}=\int -\frac{1}{x^4}dx+C\\ & \frac{y}{x^2}=\frac{1}{3x^3}+C\end{array}$ Put $f(1)=1$ $\begin{array}{rl}& C=\frac{2}{3}\\ & y=\frac{1}{3x}+\frac{2x^2}{3}\\ & y=\frac{2x^3+1}{3x}\\ & f(2)=\frac{17}{6}\\ & f(3)=\frac{55}{9}\\ & 2f(2)+3f(3)=\frac{17}{3}+\frac{55}{3}=\frac{72}{3}=24\end{array}$