We know that for large values of x, higher powers always eventually
prevail over lower powers. Thus we can claim that if x is
sufficiently large, then
Sometimes the precise solution would be difficult; then we can for instance use the notion of limit. In our example we would show that
which in particular means that there is a constant K (we may assume
that it is larger than 3) such that for all
Whatever justification you use, for
This imples that also
Since the integral on the right converges, by the Comparison test, the integral of the given function from K to infinity is also convergent. Between 3 and K there are no problems, so the given function is Riemann integrable there. Putting these two facts together we see that the given integral (from 3 to infinity) is convergent.
Now that we know that the given integral converges, we can use the failed
attempt at the Comparison test to obtain a lower estimate for its value: It
is at least