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# If the function f(x) given by $f(x) = \left\{ \begin{array}{l l}3ax+b, & \quad if { x > 0} \\ 11, & \quad if { x = 1} \\ 5ax-2b, & \quad if { x < 1 } \end{array} \right.$ is continuous at x = 1, find the values of a and b.

Can you answer this question?

Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
$f(x) = \left\{ \begin{array}{l l}3ax+b, & \quad if { x > 0} \\ 11, & \quad if { x = 1} \\ 5ax-2b, & \quad if { x < 1 } \end{array} \right.$
It is given that $f(x)$ is continuous at $x=1$
(i.e) LHL=RHL
LHL is
$\lim\limits_{x\to 1^-}f(x)=\lim\limits_{x\to 1^-}3ax+b$
$\Rightarrow 3a+b=11$
Step 2:
RHL is
$\lim\limits_{x\to 1^+}f(x)=\lim\limits_{x\to 1^+}5ax-2b$
$\Rightarrow 5a-2b=11$
Step 3:
But LHL = RHL
$3a+b=11$
$5a-2b=11$
On solving we get,
$6a+2b=22$
$5a-2b=11$
________________
$a=11$
$b=-22$
Hence a=11 and b=-22
answered Dec 2, 2013