#### Provide solution for RD Sharma maths class 12 chapter 8 Continuity exercise multiple choice question 34

The correct option is (b)

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 a} \frac{sin\: x}{x}=1 \\ \end{aligned}

Given:

$f(x)= \begin{cases}\frac{1-\cos 10 x}{x^{2}} &, x<0 \\ a & , x=0 \\ \frac{\sqrt{x}}{\sqrt{625+x}-25} &, x>0\end{cases}$

then the value of a so that f(x) may be continuous at x = 0

Using LHL

\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(-h) \\ &\quad=\lim _{h \rightarrow 0} \frac{1-\cos (-10 h)}{(-h)^{2}} \\ &\quad=50 \lim _{h \rightarrow 0} \frac{(\sin 5 h)^{2}}{(5 h)^{2}} \\ &\quad=50(\text { Using standard limits }) \end{aligned}

Function f(x) is continuous at x=0

50 = a

So, the correct option is (b).