If given function is continuous at x=1, then find a and b. f(x)= … - Vidyadip

Question

If given function is continuous at $x=1$, then find $a$ and $b. f(x)=\left\{\begin{array}{cl} 3 a x+b, & \text { if } x>1 \\ 11, & \text { if } x=1 \\ 5 a x-2 b, & \text { if } x<1 \end{array}\right. $

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Answer

Given function
$ f(x)=\left\{\begin{array}{cl} 3 a x+b, & \text { if } x>1 \\ 11, & \text { if } x=1 \\ 5 a x-2 b, & \text { if } x<1 \end{array}\right. $
value of $\text{L.H.L.}$
$\lim _{x \rightarrow 1^{-}} f(x) =\lim _{x \rightarrow 1^{-}}[5 a x-2 b]$
$ =\lim _{h \rightarrow 0}[5 a(1-h)-2 b]$
$ =\lim _{h \rightarrow 0}[5 a-2 b-5 a h]$
$ =5 a-2 b$
$\therefore \lim _{x \rightarrow 1^{-}} f(x) =5 a-2 b .$
value of $\text{R.H.L.}$
$=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(3 a x+b)$
$ =\lim _{h \rightarrow 0}[3 a(1+h)+b]$
$ =\lim _{h \rightarrow 0}(3 a+b+3 a h)=3 a+b$
$\therefore \lim _{x \rightarrow l^{+}} =3 a+b$
and $f(1) =11$
because function is continuous at $x=1$
$ f(1) =\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0} f(1-h)$
$\Rightarrow 11=3 a+b=5 a-2 b$
$\Rightarrow 5 a-2 b=11....(1)$
and $3 a+b=11....(2)$
Solving equation $(1)$ and $(2)$
hence $ a=3, b=2$

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