If(x) = {mx2 + n, x < 0
mx + m, 0 ≤ x ≤ 1
nx2 + m, x > 1
For what integers m and n does both limx→0 f(x) and limx→1 f(x) exist?
limx→0- f(x) = limx→0- (mx2 + n) = m(0)2 + n = n
limx→0+ f(x) = limx→0+ (nx + m) = n(0) + m = m
∴ limx→0 f(x) exists if m = n
limx→1- f(x) = limx→1- (nx + m) = n(1) + m = m + n
limx→1+ f(x) = limx→1+ (nx3 + m) = n(1)3 + m = m + n
∴ limx→1 f(x) exists for any integral values of m and n