SATHEE: Limit Continuity and Differentiability 5 Question 4

Limit Continuity and Differentiability 5 Question 4

4. If $f (x) = [x] - \frac{x}{4}, x \in R$ where $[x]$ denotes the greatest integer function, then

(a) $lim_{x \to 4 +} f (x)$ exists but $lim_{x \to 4 -} f (x)$ does not exist

(b) $f$ is continuous at $x = 4$

(d) $lim_{x \to 4 -} f (x)$ exists but $lim_{x \to 4 +} f (x)$ does not exist

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Answer:

Correct Answer: 4. (d)

Solution:

Given, $f (x) = \begin{array}{ll} 1 + x, & 0 \leq x \leq 2 3 - x, & 2 < x \leq 3 \end{array}$

$\begin{aligned} ∴ f \circ f (x) = f [f (x)] = \begin{array}{ll} 1 + f (x), & 0 \leq f (x) \leq 2 \\ 3 - f (x), & 2 < f (x) \leq 3 \end{array} \\ 1 + f (x), 0 \leq f (x) \leq 1 1 + (3 - x), 2 < x \leq 3 \\ \Rightarrow f \circ f = 1 + f (x), 1 < f (x) \leq 2 = 1 + (1 + x), 0 \leq x \leq 1 \\ 3 - f (x), 2 < f (x) \leq 3 3 - (1 + x), 1 < x \leq 2 \\ 4 - x, 2 < x \leq 3 \\ \Rightarrow (f \circ f) (x) = 2 + x, 0 \leq x \leq 1 \\ 2 - x, 1 < x \leq 2 \end{aligned}$

Now, $RHL ($ at $x = 2) = 2$ and LHL $($ at $x = 2) = 0$

Also, RHL $($ at $x = 1) = 1$ and LHL (at $x = 1) = 3$

Therefore, $f (x)$ is discontinuous at $x = 1, 2$

$∴ f [f (x)]$ is discontinuous at $x = 1, 2$ .

Limit Continuity and Differentiability 5 Question 4

4. If $f (x) = [x] - \frac{x}{4}, x \in R$ where $[x]$ denotes the greatest integer function, then

Answer:

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Limit Continuity and Differentiability 5 Question 4

4. If f(x)=[x]−x4,x∈R where [x] denotes the greatest integer function, then

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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4. If $f (x) = [x] - \frac{x}{4}, x \in R$ where $[x]$ denotes the greatest integer function, then