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  • Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 10 Subquestion (vii)

Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 10 Subquestion (vii)

Answers (1)

Answer:

                Discontinuous

Hint:

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Solution:

Given,

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

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

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

We observe

[LHL at x=0 ]

                \lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0}(-h-2)=-2

[RHL at x=0 ]

                \begin{aligned} &\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(h)=\lim _{h \rightarrow 0}(h+2)=2 \\\\ &\lim _{x \rightarrow 0^{+}} f(x) \neq \lim _{x \rightarrow 0^{-}} f(x) \end{aligned}

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

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