#### Need solution for RD Sharma maths class 12 chapter 8 Continuity exercise multiple choice question 35

The correct option is (a)

Hint:

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

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

(ii) Standard limits

\begin{aligned} &\lim _{x \rightarrow 0} \frac{\sin x}{x}=1 \\ &\text { (iii) } \lim _{x \rightarrow 0}\{f(x) \pm g(x)\}=1 \pm m \text {, } \end{aligned}

\begin{aligned} &{\text { Where, }} \lim _{x \rightarrow a} f(x)=1, \lim _{x \rightarrow a} g(x)=m \\ \end{aligned}

Given:

\begin{aligned} & f(x)=x \sin \frac{1}{x} \end{aligned}

Solution:

\begin{aligned} & f(x)=x \sin \frac{1}{x} \end{aligned}

$=\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} x \sin \frac{1}{x}=0$

Function f(x) is continuous at x = 0

\begin{aligned} &=\lim _{x \rightarrow 0} f(x)=f(0) \\ &=0=f(0) \end{aligned}

So, option (a) is correct.