Mathematics - Limits Question with Solution | TestHub

${\lim }_{x\to 0}2\left(\frac{1-\left(1-\frac{x^2}{2!}\right)\left(1-\frac{4x^2}{2!}\right)\left(1-\frac{9x^2}{2!}\right)\dots .\left(1-\frac{100x^2}{2!}\right)}{x^2}\right)$
By expansion $\begin{array}{c}\underset{x\to 0}{\lim }\frac{2\left(1-\left(1-\frac{x^2}{2}\right)\right)\left(1-\frac{1}{2}\cdot \frac{4x^2}{2}\right)\left(1-\frac{1}{3}\cdot \frac{9x^2}{2}\right)\dots ..\left(1-\frac{1}{10}\cdot \frac{100x^2}{2}\right)}{x^2})\\ \underset{x\to 0}{\lim }2\left(\frac{1-\left(1-\frac{x^2}{2}\right)\left(1-\frac{2x^2}{2}\right)\left(1-\frac{3x^2}{2}\right)\dots ..\left(1-\frac{10x^2}{2}\right))}{x^2}\right)\\ \underset{x\to 0}{\lim }\frac{2\left(1-1+x^2\left(\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\dots .+\frac{10}{2}\right)\right)}{x^2}\\ 2\left(\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\dots .+\frac{10}{2}\right)\\ 1+2+\dots \dots +10=\frac{10\times 11}{2}=55\end{array}$