SATHEE: Limit Continuity and Differentiability 3 Question 3

Limit Continuity and Differentiability 3 Question 3

3. $lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r}{\sqrt{n^{2} + r^{2}}}$ equals

$(1999, 2 M)$

(a) $1 + \sqrt{5}$

(b) $\sqrt{5} - 1$

(d) $1 + \sqrt{2}$

$(2007, 3 M)$

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Answer:

Correct Answer: 3. (a)

Solution:

Given, $f (x) = [\tan^{2} x]$

Now, $- 45^{\circ} < x < 45^{\circ}$

$\begin{array}{lc} \Rightarrow & \tan (- 45^{\circ}) < \tan x < \tan (45^{\circ}) \\ \Rightarrow & - \tan 45^{\circ} < \tan x < \tan (45^{\circ}) \\ \Rightarrow & - 1 < \tan x < 1 \\ \Rightarrow & 0 < \tan^{2} x < 1 \\ \Rightarrow & [\tan^{2} x] = 0 \end{array}$

i.e. $f (x)$ is zero for all values of $x$ from $x = - 45^{\circ}$ to $45^{\circ}$ . Thus, $f (x)$ exists when $x \to 0$ and also it is continuous at $x = 0$ . Also, $f (x)$ is differentiable at $x = 0$ and has a value of zero.

Therefore, (b) is the answer.

Limit Continuity and Differentiability 3 Question 3

3. $lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r}{\sqrt{n^{2} + r^{2}}}$ equals

Answer:

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Limit Continuity and Differentiability 3 Question 3

3. limn→∞1n∑r=12nrn2+r2 equals

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3. $lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r}{\sqrt{n^{2} + r^{2}}}$ equals