Continuity Problems?

Question

1. Let f be a function defined by f(x){ |x-1| + 2, for x < 1, ax^2 + bx, for x (greater than or equal to) 1 where a and b are constants.
a. If a = 2 and b = 3, is f continuous for all x?
b. Describe all values of a and b for which f is a continuous function.

2. Let f be a function defined by f(x) = (2x-2)/(x^2 + x - 2).
a. For what values of x is f discontinuous?
b. At each point of discontinuity found in part(a), determine whether f(x) has a limit, and, if so, give the value of the limit.

Answer 1

1. a. If a =2 and b = 3, then lim(x -> 1-) (f(x)) = 2, and lim(x -> 1+) = 5. Since left- and right-hand limits differ, the function is not continuous at 1.

b. We must choose a and b such that lim (x -> 1+) = 2. So we want ax^2 + bx = 2. Thus, we want x = (-b +/- sqrt(b^2 + 8a))/2a = 1. That tells us that b = (2 - a)/a. Clearly, there are lots of pairs a,b.