SATHEE: Limit Continuity and Differentiability 4 Question 7

Limit Continuity and Differentiability 4 Question 7

7. Let $[x]$ be the greatest integer less than or equals to $x$ . Then, at which of the following point(s) the function $f (x) = x \cos (π (x + [x]))$ is discontinuous? (2017 Adv.)

(a) $x = - 1$

(b) $x = 1$

(d) $x = 2$

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Answer:

Correct Answer: 7. $(a, b, d)$

Solution:

Given, $f (x) = \frac{1}{2} x - 1$ for $0 \leq x \leq π$

$\begin{aligned} ∴ [f (x)] = \begin{array}{cc} - 1, & 0 \leq x < 2 \\ 0, & 2 \leq x \leq π \end{array} \\ \Rightarrow \tan [f (x)] = \begin{array}{cc} \tan (- 1), & 0 \leq x < 2 \\ \tan 0, & 2 \leq x \leq π \end{array} \\ ∴ lim_{x \to 2^{-}} \tan [f (x)] = - \tan 1 \end{aligned}$

and $lim_{x \to 2^{+}} \tan [f (x)] = 0$

So, $\tan f (x)$ is not continuous at $x = 2$ .

Now, $f (x) = \frac{1}{2} x - 1 \Rightarrow f (x) = \frac{x - 2}{2} \Rightarrow \frac{1}{f (x)} = \frac{2}{x - 2}$

Clearly, $1 / f (x)$ is not continuous at $x = 2$ .

So, $\tan [f (x)]$ and $\frac{1}{f (x)}$ are both discontinuous at $x = 2$ .

Limit Continuity and Differentiability 4 Question 7

7. Let $[x]$ be the greatest integer less than or equals to $x$ . Then, at which of the following point(s) the function $f (x) = x \cos (π (x + [x]))$ is discontinuous? (2017 Adv.)

Answer:

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Limit Continuity and Differentiability 4 Question 7

7. Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f(x)=xcos⁡(π(x+[x])) is discontinuous? (2017 Adv.)

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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7. Let $[x]$ be the greatest integer less than or equals to $x$ . Then, at which of the following point(s) the function $f (x) = x \cos (π (x + [x]))$ is discontinuous? (2017 Adv.)