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

$a=3, b=-8$

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

Solution:

$f(x)= \begin{cases}1 & , \text { if } x \leq 3 \\ a x+b, & \text { if } 3

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

$a=3, b=-8$