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

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Answer:

Correct Answer: 3. (b)

Solution:

  1. Let $I=\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=1}^{2 n} \frac{r}{\sqrt{n^{2}+r^{2}}}=\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=1}^{2 n} \frac{r}{n \sqrt{1+(r / n)^{2}}}$

$=\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=1}^{2 n} \frac{r / n}{\sqrt{1+(r / n)^{2}}}$

$=\int _0^{2} \frac{x}{\sqrt{1+x^{2}}} d x=\left[\sqrt{1+x^{2}}\right] _0^{2}=\sqrt{5}-1$



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