Question: test for convergence

Let's consider the following functions:

1/x,1/(exp(x)-1),1/sqrt(x)-all have a singularity at x=0.All got almost similar plot.Now,making the decision if the integration blows up or not at x=0,just by looking at the plot could be misleading.So after integration from 0 to a finite value,say 1, Maple provides the answer-the first two diverge,the last one converges to 2.Now how can I be certain that Maple just simply didn't make a mistake about the first two?Is there any Maple function/procedure that can test the convergence of the integration of a function at a particular point? 

I know that the first two really are divergent integrals-it's abcd of calculus,but when I have to work with a complicated function,it is good to know for sure that Maple isn't making mistakes. 

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