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  • Provide solution for RD Sharma maths class 12 chapter Continuity exercise 8.2 question 17

Provide solution for RD Sharma maths class 12 chapter Continuity exercise 8.2 question 17

Answers (1)

Answer:

 f(x) is continuous.

Hint:

 Check at x = 0

Given:

 f(x)=\left\{\begin{array}{cc} x^{2} \sin 1 / x & x \neq 0 \\ 0 & x=0 \end{array}\right.

Explanation:

f(x)=\left\{\begin{array}{cc} x^{2} \sin 1 / x & x \neq 0 \\ 0 & x=0 \end{array}\right.

f is defined at all points of the real line

Let c be a real number

Case 1:

\begin{aligned} &\text { If } c \neq 0 \text { then } \mathrm{f}(c)=c^{2} \sin \frac{1}{c}\\ &\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left(x^{2} \sin \frac{1}{x}\right)=\lim _{x \rightarrow c}\left(x^{2}\right) \cdot \lim _{x \rightarrow c}\left(\sin \frac{1}{x}\right)\\ \end{aligned}

\begin{aligned} &=c^{2} \sin \frac{1}{c}\\ &\therefore \lim _{x \rightarrow c} f(x)=f(c)\\ &\therefore f \text { is continious at all points } \mathrm{x} \neq 0 \end{aligned}

Case 2:

\begin{aligned} &\text { If } \mathrm{c}=0 \text { then } \mathrm{f}(0)=0\\ &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}\left(x^{2} \sin \frac{1}{x}\right)\\ &\text { It is known that }-1 \leq \sin \frac{1}{x} \leq 1, x \neq 0\\ &\Rightarrow-x^{2} \leq x^{2} \sin \frac{1}{x} \leq x^{2} \end{aligned}

\begin{aligned} &\Rightarrow-x^{2} \leq x^{2} \sin \frac{1}{x} \leq x^{2} \\ &\Rightarrow \lim _{x \rightarrow 0}\left(-x^{2}\right) \leq \lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right) \leq \lim _{x \rightarrow 0} x^{2} \end{aligned}

\begin{aligned} &\Rightarrow 0 \leq \lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right) \leq 0 \\ &\Rightarrow \lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right)=0 \\ &\therefore \lim _{x \rightarrow 0^{-}} f(x)=0 \end{aligned}

Similarly,

\begin{aligned} &\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}\left(x^{2} \sin \frac{1}{x}\right) \\ &=\lim _{x \rightarrow 0}\left(x^{2} \sin \frac{1}{x}\right)=0 \\ &\therefore \lim _{x \rightarrow 0^{-}} f(x)=f(0)=\lim _{x \rightarrow 0^{+}} f(x) \end{aligned}

Therefore f is continuous at x=0

From the above observations.

It can be conclude that f is continuous at every point of the real line.

Thus f is a continuous function.

Posted by

Gurleen Kaur

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