MCQ
$\mathop {\lim }\limits_{x \to 0} \,x^2(1+2+3+...+[\frac{1}{|x|}])$ is equal to
(where [.] denotes greatest integer function)
- A$0$
- ✓$\frac{1}{2}$
- C$2$
- Ddoes not exist
(where [.] denotes greatest integer function)
$\therefore $ Given limit $ = \mathop {\lim }\limits_{t \to \infty } \frac{{1 + 2 + 3 + .... + \left[ {\left| t \right|} \right]}}{{{t^2}}}$
$ = \mathop {\lim }\limits_{t \to \infty } \frac{{[|t|]([|t|] + 1)}}{{2{t^2}}} = \frac{1}{2}$
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