Mathematics - Limits Question with Solution | TestHub

$\begin{array}{rl}& \underset{x\to \frac{1}{a}}{\lim }16\cdot \frac{\left(1-\cos 2\left(x-\frac{1}{a}\right)\left(x-\frac{1}{b}\right)\right)}{4{\left(x-\frac{1}{b}\right)}^2}\times \frac{4{\left(x-\frac{1}{b}\right)}^2}{a^2{\left(x-\frac{1}{a}\right)}^2}\\ & =16\times \frac{2}{a^2}{\left(\frac{1}{a}-\frac{1}{b}\right)}^2\\ & =\frac{32}{a^2}\left(\frac{17}{4}\right)=\frac{17.8}{a^2}=\frac{17\times 8\times 16}{(-1+\sqrt{117}{)}^2}\\ & =\frac{136.16}{18.2\sqrt{7}}\times \frac{18+2\sqrt{7}}{18+2\sqrt{7}}\\ & =\frac{136}{256}(18+2\sqrt{7})\cdot 16\\ & =153+17\sqrt{17}=\alpha +\beta \sqrt{17}\\ & \alpha +\beta =153+17=170\end{array}$