SATHEE: Limit Continuity and Differentiability 4 Question 3

Limit Continuity and Differentiability 4 Question 3

3. Let [.] denotes the greatest integer function and $f (x) = [\tan^{2} x]$ , then

(1993, 1M)

(a) $lim_{x \to 0} f (x)$ does not exist

(b) $f (x)$ is continuous at $x = 0$

(d) $f^{'} (0) = 1$

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Answer:

Correct Answer: 3. (b)

Solution:

Given function

$f (x) = \begin{array}{ll} a | π - x | + 1, & x \leq 5 \\ b | x - π | + 3, & x > 5 \end{array}$

and it is also given that $f (x)$ is continuous at

Clearly,

$f (5) = a (5 - π) + 1$

$\begin{aligned} lim_{x \to 5^{-}} f (x) & = lim_{h \to 0} [a | π - (5 - h) | + 1] \\ = a (5 - π) + 1 \end{aligned}$

and $lim_{x \to 5^{+}} f (x) = lim_{h \to 0} [b | (5 + h) - π | + 3]$

$= b (5 - π) + 3$

$∵$ Function $f (x)$ is continuous at $x = 5$ .

$\begin{aligned} ∴ f (5) = lim_{x \to 5^{+}} f (x) = lim_{x \to 5^{-}} f (x) \\ \Rightarrow a (5 - π) + 1 = b (5 - π) + 3 \\ \Rightarrow (a - b) (5 - π) = 2 \\ \Rightarrow a - b = \frac{2}{5 - π} \end{aligned}$

Limit Continuity and Differentiability 4 Question 3

3. Let [.] denotes the greatest integer function and $f (x) = [\tan^{2} x]$ , then

Answer:

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Limit Continuity and Differentiability 4 Question 3

3. Let [.] denotes the greatest integer function and f(x)=[tan2⁡x], then

Answer:

Engineering Preparation

Medical Preparation

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Syllabus

Test Platform

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3. Let [.] denotes the greatest integer function and $f (x) = [\tan^{2} x]$ , then