Limit Continuity and Differentiability 2 Question 3
3. Let $[x]$ denote the greatest integer less than or equal to $x$. Then,
$$ \lim _{x \rightarrow 0} \frac{\tan \left(\pi \sin ^{2} x\right)+(|x|-\sin (x[x]))^{2}}{x^{2}} $$
(a) equals $\pi$
(b) equals $\pi+1$
(c) equals 0
(d) does not exist
Show Answer
Answer:
Correct Answer: 3. (b)
Solution:
- Let $I=\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=1}^{2 n} \frac{r}{\sqrt{n^{2}+r^{2}}}=\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=1}^{2 n} \frac{r}{n \sqrt{1+(r / n)^{2}}}$
$=\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=1}^{2 n} \frac{r / n}{\sqrt{1+(r / n)^{2}}}$
$=\int _0^{2} \frac{x}{\sqrt{1+x^{2}}} d x=\left[\sqrt{1+x^{2}}\right] _0^{2}=\sqrt{5}-1$
