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Question

Express a in terms of b if the function f defined by $$f(x)=\begin{Bmatrix} ax+1 & , x \leq 3 \\ bx+3 & x>3 \end{Bmatrix}$$ is continuous at $$x=3$$.


Solution

$$f\left( x \right) =\left\{ \begin{matrix} ax+1,\quad x\le 3 \\ bx+3\quad x>3 \end{matrix} \right. at\ x=3$$

function is continuous at $$x=3$$

$$LHL=RHL$$

$$\displaystyle \lim_{x\rightarrow 3^-}{(ax+1)}=\displaystyle \lim_{x\rightarrow 3^+}{(bx+3)}$$

$$a(3)+1=b(3)+3$$

$$3a=3b+2$$

$$\boxed{a=b+\dfrac23}$$

Mathematics

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