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  • Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 1

Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 1

Answers (1)

Answer:

                Discontinuous

Hint:

The discontinuity occurs when the LHL and RHL are not equal at a point

Solution:

Given

                f(x)=\left(\begin{array}{l} \frac{x}{|x|}, x \neq 0 \\\\ 1, x=0 \end{array}\right)

We observe,

[LHL at x=0 ]

                \begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h) \\ &\lim _{h \rightarrow 0} \frac{-h}{|-h|}=\lim _{h \rightarrow 0} \frac{-h}{h}=\lim _{h \rightarrow 0}(-1)=-1 \end{aligned}

[RHL at x=0 ]

\therefore          \begin{aligned} &\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0} f(h) \\ &\lim _{h \rightarrow 0} \frac{h}{|h|}=\lim _{h \rightarrow 0} \frac{h}{h}=\lim _{h \rightarrow 0}(1)=1 \\ &\lim _{x \rightarrow 0^{+}} f(x) \neq \lim _{x \rightarrow 0^{-}} f(x) \end{aligned}

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

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