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  • Provide Solutio for RD Sharma Class 12 Chapter Continuity Exercise 8.1 Question 36 Subquestion (iv)

Provide Solutio for RD Sharma Class 12 Chapter Continuity Exercise 8.1 Question 36 Subquestion (iv)

Answers (1)

Answer:

                k=\frac{-2}{\pi}

Hint:

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

 

Given:

                f(x)=\left\{\begin{array}{l} k x+1, \text { if } x \leq \pi \\\\ \cos x, \text { if } x>\pi \end{array}\right.

               

Solution:

                f(x)=\left\{\begin{array}{l} k x+1, \text { if } x \leq \pi \\\\ \cos x, \text { if } x>\pi \end{array}\right.

We have

(LHL at x=\pi )

                \lim _{x \rightarrow \pi^{-}} f(x)=\lim _{h \rightarrow 0} f(\pi-h)=\lim _{h \rightarrow 0} k(\pi-h)+1=k \pi+1

(RHL at x=\pi )

                \lim _{x \rightarrow \pi^{+}} f(x)=\lim _{h \rightarrow 0} f(\pi+h)=\lim _{h \rightarrow 0} \cos (\pi+h)=\cos \pi=-1

If f\left ( x \right ) is continuous at x=\pi , then

                \begin{aligned} &\lim _{x \rightarrow \pi^{+}} f(x)=\lim _{x \rightarrow \pi^{-}} f(x) \\ &k \pi+1=-1 \\ &k=\frac{-2}{\pi} \end{aligned}

               

 

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