Mathematics - Application of Derivative Question with Solution | TestHub

slope of $3x+2y=1$ is $\frac{-3}{2}\Rightarrow f^{\prime }\left(\frac{1}{2}\right)=\frac{-3}{2}$ $\begin{array}{rlr}& p=\underset{x\to 0}{\lim }\frac{x(x-1)}{f\left(\frac{e^{2x}}{2}\right)-f\left(\frac{e^{-2x}}{2}\right)}\left(\frac{0}{0}\right)=\frac{-1}{f^{\prime }\left(\frac{1}{2}\right)+f^{\prime }\left(\frac{1}{2}\right)}=\frac{1}{3}& \therefore \sum _{r=1}^{\infty }{p}^r=\frac{1}{3}+\frac{1}{3^2}+\dots \infty =\frac{1/3}{1-1/3}=\frac{1/3}{2/3}=\frac{1}{2}\end{array}$