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# Find the relationship between 'a' and 'b' so that the function 'f' defined by $f(x) = \left\{ \begin{array}{l l}ax+1, & \quad if { x \leq 3} \\ bx+3, & \quad if { x > 3} \end{array} \right.$ is continuous at x = 3.

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A)
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:
At $x=3$
LHL=$\lim\limits_{\large x\to 3^-}(ax+1)$
$\quad\quad=3a+1.$
$f(3)=3a+1$
RHL=$\lim\limits_{\large x\to 3^+}(bx+3)$
$\quad\quad=3b+3.$
$f$ is continuous if LHL=RHL=f(3)
Step 2:
$3a+1=3b+3$
$3a-3b=3-1$
$3a-3b=2$
$3(a-b)=2$
$a-b=\large\frac{2}{3}$
$a=\large\frac{2}{3}$$+b$
For any arbitrary value of $b$ we can find the value of $a$ corresponding the value of $b$.