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  • Please solve RD Sharma class 12 chapter 8 Continuity exercise multiple choice question 17 maths textbook solution

Please solve RD Sharma class 12 chapter 8 Continuity exercise multiple choice question 17 maths textbook solution

Answers (1)

Answer:

 The correct option is (c)

Hint:

 A function f(x) is said to be continuous at a point x = a of its domain, if

\lim _{x \rightarrow a^{+}} f(a+h)=\lim _{x \rightarrow a^{-}} f(a-h)=f(a)

Given:

 f(x)=\left\{\begin{array}{l} 1,|x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}<|x|<\frac{1}{n-1}, n=2,3, \ldots \\ 0, x=0 \end{array}\right.

Solution:

Step 1: Simplify the given function

f(x)=\left\{\begin{array}{l} 1,|x| \geq 1 \\ \frac{1}{n^{2}}, \frac{1}{n}<|x|<\frac{1}{n-1}, n=2,3, \ldots \\ 0, x=0 \end{array}\right.

Case 2: Calculate the RHL

\begin{gathered} \lim _{x \rightarrow \frac{1}{n}^{+}} f(x)=\lim _{h \rightarrow 0}\left(\frac{1}{n}+h\right) \\ =\left(\frac{1}{h}\right)^{2} \end{gathered}

Case 3: Calculate the RHL

\begin{aligned} &\lim _{x \rightarrow \frac{1}{n}^{-}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{1}{n}-h\right)=\left(\frac{1}{n-1}\right)^{2} \\ &\text { Therefore, } \lim _{x \rightarrow \frac{1}{n}^{+}} f(x) \neq \lim _{x \rightarrow \frac{1}{n}^{-}} f(x) \end{aligned}

Hence, the correct answer is option (c)

Hence, f(x) is discontinuous only at

x=\pm \frac{1}{n^{\prime}} n \in Z-\{0\} \text { and } x=0

Posted by

Gurleen Kaur

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