SATHEE: Limit Continuity and Differentiability 4 Question 2

Limit Continuity and Differentiability 4 Question 2

2. The function $f (x) = [x]^{2} - [x^{2}]$ (where, $[x]$ is the greatest integer less than or equal to $x$ ), is discontinuous at

(a) all integers

$(1999, 2 M)$

(b) all integers except 0 and 1

(d) all integers except 1 Objective Questions II

(One more than one correct option)

Show Answer

Answer:

Correct Answer: 2. (b)

Solution:

Given function

$f (x) = \begin{array}{cc} \frac{\sin (p + 1) x + \sin x}{x} & , x < 0 \\ q & , x = 0 \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} & , x > 0 \end{array}$

is continuous at $x = 0$ , then

$\begin{aligned} f (0) = lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{+}} f (x) \dots (i) \\ lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{-}} \frac{\sin (p + 1) x + \sin x}{x} \\ = p + 1 + 1 = p + 2 ∵ lim_{x \to 0} \frac{\sin (a x)}{x} = a \\ and lim_{x \to 0^{+}} f (x) = lim_{x \to 0^{+}} \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} \\ = lim_{x \to 0^{+}} \frac{\sqrt{x} [(1 + x)^{1 / 2} - 1]}{x \sqrt{x}} \\ 1 + \frac{1}{2} x + \frac{\frac{1}{2} \frac{1}{2} - 1}{2!} x^{2} + \dots . - 1 \\ = lim_{x \to 0^{+}} \frac{}{x} \\ [∵ (1 + x)^{n} \\ = 1 + n x + \frac{n (n - 1)}{1 \cdot 2} x^{2} + \frac{n (n - 1 (n - 2))}{1 \cdot 2 \cdot 3} x^{3} + \dots, | x | < 1] \end{aligned}$

$= lim_{x \to 0^{+}} \frac{1}{2} + \frac{\frac{1}{2} \frac{1}{2} - 1}{2!} x + \dots = \frac{1}{2}$

From Eq. (i), we get

$\begin{aligned} f (0) & = q = \frac{1}{2} and lim_{x \to 0^{-}} f (x) = p + 2 = \frac{1}{2} \\ \Rightarrow & p & = - \frac{3}{2} \\ So, & (p, q) & = - \frac{3}{2}, \frac{1}{2} \end{aligned}$

Limit Continuity and Differentiability 4 Question 2

2. The function $f (x) = [x]^{2} - [x^{2}]$ (where, $[x]$ is the greatest integer less than or equal to $x$ ), is discontinuous at

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Limit Continuity and Differentiability 4 Question 2

2. The function f(x)=[x]2−[x2] (where, [x] is the greatest integer less than or equal to x ), is discontinuous at

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

2. The function $f (x) = [x]^{2} - [x^{2}]$ (where, $[x]$ is the greatest integer less than or equal to $x$ ), is discontinuous at