Evaluate $ \lim _{x \rightarrow 0}\left(x^{2}\left(1+2+3+\dots+\left[\frac{1}{|x|}\right]\right)\right) $

Question

Evaluate $$ \lim _{x \rightarrow 0}\left(x^{2}\left(1+2+3+\dots+\left[\frac{1}{|x|}\right]\right)\right) $$ For any real number $a,|a|$ is the largest integer not greater than $a$.

I am getting no clue! from where to start!

Answer 1

We have \begin{align*} & \mathop {\lim }\limits_{x \to 0} \left( {x^2 \left( {1 + 2 + 3 + \cdots + \left[ {\frac{1}{{\left| x \right|}}} \right]} \right)} \right) = \mathop {\lim }\limits_{x \to 0} \left( {\left| x \right|^2 \left( {1 + 2 + 3 + \cdots + \left[ {\frac{1}{{\left| x \right|}}} \right]} \right)} \right) \\ & = \mathop {\lim }\limits_{n \to + \infty } \left( {\frac{1}{{n^2 }}\left( {1 + 2 + 3 + \cdots + n} \right)} \right) = \mathop {\lim }\limits_{n \to + \infty } \frac{{n(n + 1)}}{{2n^2 }} = \frac{1}{2}. \end{align*}

Answer 2

The sum of the first $n$ integers is $\frac{n(n+1)}{2}$. Now, we can express $n$ with the limit of $\big \lfloor \frac{1}{|x|} \big \rfloor$. $$\lim_{x \to 0} x^2 \frac{\big \lfloor \frac{1}{|x|} \big \rfloor \left(\big \lfloor \frac{1}{|x|} \big \rfloor+1\right)}{2}=\frac{1}{2}$$