Question

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

Answer 1

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)(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