Question
Find the relationship between a and b, so that the function f defined by f(x)= $ \left\{ \begin{array} { l } { a x + 1 , \text { if } x \leq 3 } \\ { b x + 3 , \text { if } x > 3 } \end{array} \right.$is continuous at x = 3.
Therefore , LHL = RHL = f(3).........(i)
Now, LHL = $ \mathop {\lim }\limits_{ x \rightarrow 3 ^ { - } } f ( x ) = \mathop {\lim }\limits_{ x \rightarrow 3 ^ { - } } ( a x + 1 )$
$ = \mathop {\lim }\limits_{ h \rightarrow 0 } [ a ( 3 - h ) + 1 ]$
$ \Rightarrow$ LHL = 3a + 1
and RHL = $ \mathop {\lim }\limits_{ x \rightarrow 3 ^ { + } } f ( x ) = \mathop {\lim }\limits_{ x \rightarrow 3 ^ { + } } ( b x + 3 )$
$ = \mathop {\lim }\limits_{ h \rightarrow 0 } [ b ( 3 + h + 3 ]$
$ = \mathop {\lim }\limits_{ h \rightarrow 0 } ( 3 b + b h + 3 )$
$ \Rightarrow$ RHL = 3b + 3
From Eq. (i), we have
LHL = RHL$ \Rightarrow$ 3a + 1 = 3b + 3
Therefore, 3a - 3b = 2, which is the required relation between a and b.
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