# 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?

Yes, it is possible for an integral in the form ∫ₐ^∞ f(x) dx, where limₓ→∞ f(x) ≠ 0, to still be convergent if the function f(x) decreases fast enough as x approaches infinity. This can occur when f(x) approaches zero at a rate that compensates for its divergence to infinity, allowing the integral to converge. This behavior is often observed in functions with exponential decay or other rapidly decreasing functions.

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So:

and based on a well known inequality:

Can't find a counterexample right now, though.

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