Solve bx + c = 0 for x by means of the quadratic formula.
(In reply to By limits
by Old Original Oskar!)
One nitpick, the extaneous solution does not even conclusively go to infinity.
Taking the limit from the positive side of 0 gives positive infinity and taking the limit from the negative side of 0 gives negative infinity. Since the limits on each side do not converge to the same value, the limit does not even exist.