_ x 0 ,x^2(1+2+3+...+[1 |x| ]) is equal to (where [.] denotes gre… - Vidyadip

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
$2$
D
does not exist

✓

Answer

Correct option: B.

$\frac{1}{2}$

b
Put $x=\frac{1}{t}$

$\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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