converges or diverges, and find its value if it converges.
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
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.