SATHEE: Limit Continuity and Differentiability 2 Question 3

Limit Continuity and Differentiability 2 Question 3

3. Let $[x]$ denote the greatest integer less than or equal to $x$ . Then,

$lim_{x \to 0} \frac{\tan (π \sin^{2} x) + (| x | - \sin (x [x]))^{2}}{x^{2}}$

(a) equals $π$

(b) equals $π + 1$

(c) equals 0

(d) does not exist

Show Answer

Answer:

Correct Answer: 3. (b)

Solution:

Let $I = lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r}{\sqrt{n^{2} + r^{2}}} = lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r}{n \sqrt{1 + (r / n)^{2}}}$

$= lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r / n}{\sqrt{1 + (r / n)^{2}}}$

$= \int_{0}^{2} \frac{x}{\sqrt{1 + x^{2}}} d x = {[\sqrt{1 + x^{2}}]}_{0}^{2} = \sqrt{5} - 1$

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Limit Continuity and Differentiability 2 Question 3

3. Let [x] denote the greatest integer less than or equal to x. Then,

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3. Let $[x]$ denote the greatest integer less than or equal to $x$ . Then,