Question

The value of $\underset{x\to 0}{lim}\sin x^2\times (\left[\frac{1}{x^2}\right]+\left[\frac{2}{x^2}\right]+\left[\frac{3}{x^2}\right]+\cdots +\left[\frac{10}{x^2}\right])$, is
(where $[.]$ denotes the greatest integer function)

Answer 1

As, $\underset{x\to 0}{lim}\left[\frac{1}{x^2}\right]\to \infty$
$\Rightarrow \underset{x\to 0}{lim}\left[\frac{1}{x^2}\right]=\underset{x\to 0}{lim}\frac{1}{x^2}$
Now,
Let $y=\underset{x\to 0}{lim}\sin x^2\times (\left[\frac{1}{x^2}\right]+\left[\frac{2}{x^2}\right]+\left[\frac{3}{x^2}\right]+\cdots +\left[\frac{10}{x^2}\right])$
becomes,
$y=\underset{x\to 0}{lim}\sin x^2(\frac{1}{x^2}+\frac{2}{x^2}+\frac{3}{x^2}+\cdots +\frac{10}{x^2})$
$\Rightarrow y=\underset{x\to 0}{lim}\frac{\sin x^2}{x^2}\times \frac{10\times 11}{2}$
$\Rightarrow y=55$