SATHEE: Limit Continuity and Differentiability 4 Question 5

Limit Continuity and Differentiability 4 Question 5

5. For each positive integer $n$ , let

$y_{n} = \frac{1}{n} ((n + 1) (n + 2) \dots (n + n))^{\frac{1}{n}}$

For $x \in R$ , let $[x]$ be the greatest integer less than or equal to $x$ . If $lim_{n \to \infty} y_{n} = L$ , then the value of $[L]$ is

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Answer:

Correct Answer: 5. (c)

Solution:

Given function $f : [- 1, 3] \to R$ is defined as

$f (x) = \begin{array}{lr} | x | + [x], & - 1 \leq x < 1 \\ x + | x |, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{array}$

$\begin{aligned} - x - 1, - 1 \leq x < 0 \\ x, 0 \leq x < 1 \\ = 2 x, 1 \leq x < 2 \\ x + 2, 2 \leq x < 3 \\ 6, x = 3 \end{aligned}$

$∴$ Function $f (x)$ is discontinuous at points 0,1 and 3 .

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Limit Continuity and Differentiability 4 Question 5

5. For each positive integer n, let

Fill in the Blank

Answer:

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5. For each positive integer $n$ , let