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  • Please Solve RD Sharma Class 12 Chapter 8 Continuity Exercise 8.1 Question 37 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 8 Continuity Exercise 8.1 Question 37 Maths Textbook Solution.

Answers (1)

Answer:

                a=3, b=-8

Hint:

f\left ( x \right )  must be defined. The limit of the f\left ( x \right ) approaches the value x must exist.

Given:

                f(x)= \begin{cases}1 & , \text { if } x \leq 3 \\ a x+b, & \text { if } 3<x<5 \\ 7 & , \text { if } x \geq 5\end{cases}

Solution:

                f(x)= \begin{cases}1 & , \text { if } x \leq 3 \\ a x+b, & \text { if } 3<x<5 \\ 7 & , \text { if } x \geq 5\end{cases}

We have

(LHL at x= 3 )

                \lim _{x \rightarrow 3^{-}} f(x)=\lim _{h \rightarrow 0} f(3-h)=\lim _{h \rightarrow 0}(1)=1

(RHL at x= 3 )

                \lim _{x \rightarrow 3^{+}} f(x)=\lim _{h \rightarrow 0} f(3+h)=\lim _{h \rightarrow 0} a(3+h)+b=3 a+b

(LHL at x= 5 )

                \lim _{x \rightarrow 5^{-}} f(x)=\lim _{h \rightarrow 0} f(5-h)=\lim _{h \rightarrow 0}[a(5-h)+b]=5 a+b

(RHL at x= 5 )

                \lim _{x \rightarrow 5^{+}} f(x)=\lim _{h \rightarrow 0} f(5+h)=\lim _{h \rightarrow 0} 7=7

If f\left ( x \right )  is continuous at x= 3  and x= 5 , then

\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3^{j}} f(x) \text { and } \lim _{x \rightarrow 5^{+}} f(x)=\lim _{x \rightarrow 5^{-}} f(x)

1=3 a+b                                                                                                                                                         … (i)

5 a+b=7                                                                                                                                                                    … (ii)

On solving equation (i) and (ii), we get

                a=3, b=-8

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