Repeated Roots, Long-Term Behavior: Show that in the "repeated roots" case of equation (1), the solution, which is given by x(t) = c1e-(b/2a)t + c2te-(b/2a)t , for b/2a > 0, tends toward zero as t becomes large. You may need l'Hpital's Rule: If, as x approaches
a. both f(x) and g(x) approach zero, then the limx†’a f(x)/g(x) is indeterminate. But Marquis l'Hpital (1661 - 1704) came to the rescue by publishing a result of Johann Bernoulli (1667-1748), that we can find the limit of the quotient by

providing, of course, that both derivatives exist in a neighborhood of a and approach nonzero limits.

f'(x) 5 f(x) lim