_ 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}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If the curve $x^{2}+2 y^{2}=2$ intersects the line $x + y =1$ at two points $P$ and $Q ,$ then the angle subtended by the line segment $PQ$ at the origin is ...... .

If in a geometric progression $\left\{ {{a_n}} \right\},\;{a_1} = 3,\;{a_n} = 96$ and ${S_n} = 189$ then the value of $n$ is

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left[\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right]$ on the hyperbola, is equal to

Let $R$ be a relation on the set $N$ be defined by $\{(x, y)| x, y \in N, 2x + y = 41\}$. Then $R$ is

Let $S=\{1,2,3, \ldots, 100\}$. Suppose $b$ and $c$ are chosen at random from the set $S$. The probability that $4 x^2+b x+c$ has equal roots is

The number that exceeds its square by the greatest amount is

For every pair of continuous functions $f, g:[0,1] \rightarrow R$ such that $\max \{f(x): x \in[0,1]\}=\max \{g(x): x \in[0,1]\}$, the correct statement$(s)$ is (are) :

$(A)$ $(f(c))^2+3 f(c)=(g(c))^2+3 g(c)$ for some $c \in[0,1]$

$(B)$ $(f(c))^2+f(c)=(g(c))^2+3 g(c)$ for some $c \in[0,1]$

$(C)$ $(f(c))^2+3 f(c)=(g(c))^2+g(c)$ for some $c \in[0,1]$

$(D)$ $(f(c))^2=(g(c))^2$ for some $c \in[0,1]$

The solution of the differential equation $(1 + {x^2})\frac{{dy}}{{dx}} = x(1 + {y^2})$ is

If $aN = \{ ax:x \in N\} $ and $bN \cap cN = dN$, where $b$, $c \in N$ are relatively prime, then

If $r\,\sin \theta = 3,r = 4(1 + \sin \theta ),\,\,0 \le \theta \le 2\pi ,$ then $\theta = $