# Is it possible to for an integral in the form #int_a^oo f(x)\ dx#, and #lim_(x->oo)f(x)!=0#, to still be convergent?

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If you view the integral as the area under the curve, it seems logical that there is no way that the integral

#int_a^oo f(x)\ dx#

would converge unless #f(x)# eventually tends to zero

#lim_(x->oo)f(x)=0#

since the area under the graph wouldn't be bounded otherwise.

My question is, are there integrals where this is not the case? Where the limit of the function doesn't go to zero, but the integral is still convergent? What would be an example of such function?

If you view the integral as the area under the curve, it seems logical that there is no way that the integral

would converge unless

since the area under the graph wouldn't be bounded otherwise.

My question is, are there integrals where this is not the case? Where the limit of the function doesn't go to zero, but the integral is still convergent? What would be an example of such function?

##### 1 Answer

If the limit

In fact suppose

So:

and based on a well known inequality:

which clearly diverges for

If

For the same reason, also if

However, if

Can't find a counterexample right now, though.