Question

determine the constraints of a and b such that g(x) is continuous for all values of x.

Answer 1

Solution

g(x) is continuous at x = 1 if

$\lim_{x\to1^-}g(x)=\lim_{x\to1+}g(x)=g(1)$$\Rightarrow\lim_{x\to1^-}g(x)=\lim_{x\to1}\frac{ax-b}{x-2}=b-a$$\lim_{x\to1^+}g(x)=\lim_{x\to1}-3x=-3$

g(x) is continuous at x = 2 if

$\lim_{x\to2^-}g(x)=\lim_{x\to2^+}g(x)=g(2)$$\Rightarrow\lim_{x\to2^-}g(x)=\lim_{x\to2}-3x=-6$$\Rightarrow\lim_{x\to2^+}g(x)=\lim_{x\to2}bx^2-a=4b-a$

So we have the equations

$\begin{gathered} b-a=-3\text{ ----\lparen1\rparen} \\ \\ 4b-a=-6\text{ ----\lparen2\rparen} \end{gathered}$

Equation (1) - (2)

$\begin{gathered} -3b=3 \\ \\ \Rightarrow b=-1 \\ \text{ Substitute b=-1 in equation \lparen1\rparen} \\ \Rightarrow a=b+3=-1+3=2 \end{gathered}$

Hence

a = 2

b = -1