# Find the values of a and b so that the function f given by

Question:

Find the values of $a$ and $b$ so that the function $f$ given by

f(x)=\left\{\begin{aligned} 1, & \text { if } x \leq 3 \\ a x+b, & \text { if } 3 Solution: Given:f(x)=\left\{\begin{aligned} 1, & \text { if } \mathrm{x} \leq 3 \\ a x+b, & \text { if } 3

We have

$(\mathrm{LHL}$ at $x=3)=\lim _{x \rightarrow 3} f(x)=\lim _{h \rightarrow 0} f(3-h)=\lim _{h \rightarrow 0}(1)=1$

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

$(\mathrm{LHL}$ at $x=5)=\lim _{x \rightarrow 5^{-}} f(x)=\lim _{h \rightarrow 0} f(5-h)=\lim _{h \rightarrow 0}(a(5-h)+b)=5 a+b$

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

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

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

$\Rightarrow 1=3 a+b \quad \ldots(1)$ and $5 a+b=7 \quad \ldots(2)$

On solving eqs. (1) and (2), we get

$a=3$ and $b=-8$