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  • provide solution for RD Sharma maths class 12 chapter 8 Continuity exercise Fill in the blanks question 2

provide solution for RD Sharma maths class 12 chapter 8 Continuity exercise  Fill in the blanks question 2

Answers (1)

Answer: \pm 1

Hint:

            Use the formula of  \lim _{x \rightarrow 0} \frac{\sin x}{x}=1  to solve \frac{\sin ^{2} a x}{x^{2}}

Given:

           f(x)=\left\{\begin{array}{cl} \frac{\sin ^{2} a x}{x^{2}}, & x \neq 0 \\ 1, & x=0 \end{array}\right.  is continuous at x=0

Solution:

            If f(x) is continuous at x=0, then f(x)=\frac{\sin ^{2} a x}{x^{2}} and g(x)=1 are equal.

            LHL,

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

            =\lim_{x\rightarrow 0^{-}}\left ( \frac{\sin ax}{ax} \right )^{2}\left ( a^{2} \right )    

            =a^{2}                                                                \left[\because \frac{\sin ^{2} a x}{a^{2} x^{2}}=1\right]

            RHL,

            \lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} 1=1

              As f(x) is continuous at x=0, LHL =RHL

             \begin{aligned} &a^{2}=1 \\ &a=\pm 1 \end{aligned}

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