Mathematics - Limits Question with Solution | TestHub

$x^2\left(1+2+3+\dots ..\left[\frac{1}{|x|}\right]\right)$

$\frac{x^2\left(1+\left[\frac{1}{|x|}\right]\right)}{2}\left[\frac{1}{|x|}\right]$

Now using the property that

$\frac{1}{|x|}-1<\left[\frac{1}{|x|}\right]\le \frac{1}{|x|}$

we get

$\frac{1}{2}|x|<\frac{x^2\left(1+\left[\frac{1}{|x|}\right]\right)}{2}\left[\frac{1}{|x|}\right]\le \frac{1}{2}(1+|x|)$

Now applying sandwich theorem the required limit is $\frac{1}{2}$