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.