Question

Suppose f(x) ={ a+bx, x$<$1 4, x=1 b-ax x$>$1 and if lim x $\rightarrow$1 f(x) = f(1), what are possible values of a and b?

Answer 1

The given function is
f(x) ={ a+bx, x<1 4, x=1 b-ax x>1
$\lim_{x\rightarrow 1^-}$ f(x) = $\lim_{x\rightarrow 1}$ (a+bx) = a+b
$\lim_{x\rightarrow 1^-}$f(x) = $\lim_{x\rightarrow 1}$ (b-ax) = b-a
f(1) = 4
It is given that $\lim_{x\rightarrow 1}$ f(x) =f(1).
∴$\lim_{x\rightarrow 1^-}$ f(x) = $\lim_{x\rightarrow 1^+}$ f(x) =$\lim_{x\rightarrow 1}$ f(x) = f(1)
$\Rightarrow$ a+b =4 and b-a =4
On solving these two equations, we obtain a =0 and b= 4.
Thus, the respective possible values of a and b are 0 and 4.