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  • Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 24

Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 24

Answers (1)

Answer:

Since,\lim _{x \rightarrow 0^{-}} f(x)   and \lim _{x \rightarrow 0^{+}} f(x)  are not equal,  f\left ( x \right )  is discontinuous.

Thus,f\left ( x \right )  is discontinuous at  x=0

Hint:

f\left ( x \right )  must be defined. The limit of the  f\left ( x \right ) approaches the value x  must exist.

Given:

          f(x)=\left\{\begin{array}{c} \frac{x}{|x|+2 x^{2}}, \text { if } x \neq 0 \\\\ k, \text { if } x=0 \end{array}\right. 

Solution:

                f(x)=\left\{\begin{array}{c} \frac{x}{x+2 x^{2}}, \text { if } x>0 \\\\ \frac{-x}{x-2 x^{2}}, \text { if } x<0 \\\\ k, \text { if } x=0 \end{array}\right.

                f(x)=\left\{\begin{array}{c} \frac{1}{2 x+1}, \text { if } x>0 \\\\ \frac{1}{2 x-1}, \text { if } x<0 \\\\ k, \text { if } x=0 \end{array}\right.

We observe

(LHL at x=0 )

                \lim _{h \rightarrow 0^{-}} \frac{1}{-2 h-1}=-1

(RHL at x=0 )

                \lim _{h \rightarrow 0^{+}} \frac{1}{2 h+1}=1

Since, \lim _{x \rightarrow 0^{-}} f(x)  and  \lim _{x \rightarrow 0^{+}} f(x) are not equal, f\left ( x \right ) is discontinuous.

 Thus, f\left ( x \right ) is discontinuous at x=0 , regardless of choice of  k

 

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