We've sort of done this problem already. We're pretty sure that this integral converges, and that it converges to 1. Let's argue the case a little more carefully. By the definition of this type of improper integral, we know that We know how to work out this integral, so we do - remembering to keep the limit notation along for the ride. We've worked out the integral inside the limit, so it's time to figure out the limit. As *b* approaches ∞, the term approaches 0. So The integral converges to 1. Remember to include the limit. If you leave a limit out, you get something like which doesn't make any sense. The left-hand side is a limit, which is either a number or doesn't exist at all. The right-hand side is just an algebraic expression with the letter *b* in it. |