Mathematics - Limits Question with Solution | TestHub

$$\underset{t\to x}{\lim }\frac{t^{2}f(x)-x^{2}f(t)}{t-x}=1$$ $$\begin{array}{rlrl}& \Rightarrow & x^2{f}^{\prime }(x)-2xf(x)+1& =0\\ & \Rightarrow & \frac{x^2{f}^{\prime }(x)-2xf(x)}{{\left(x^2\right)}^2}+\frac{1}{x^4}& =0\\ & \Rightarrow & \frac{d}{dx}\left(\frac{f(x)}{x^2}\right)& =-\frac{1}{x^4}\\ \Rightarrow & & f(x)& =c{x}^2+\frac{1}{3x}\text{ also }f(1)=1\Rightarrow c=\frac{2}{3}.\\ & \text{ Hence, }& f(x)& =\frac{2}{3}{x}^2+\frac{1}{3x}\end{array}$$