Help understanding Direct/Limit comparison tests

[IntegrateMe]

I'm having an extremely hard time understanding direct/limit comparison tests, and i will post some questions so you guys can attempt to help me.

integral from 0 to pi of: dt/[sqrt(t) + sin(t)]

and

integral from -infinity to infinity of: dx/[e^x + e^-x]

I cannot show my work because i have no idea how to do these problems. If someone can explain the direct/limit comparison tests then i will attempt these problems and post my work so you guys can guide me. Thanks.

 

[LCKurtz]

I'm having an extremely hard time understanding direct/limit comparison tests, and i will post some questions so you guys can attempt to help me.

integral from 0 to pi of: dt/[sqrt(t) + sin(t)]

and

integral from -infinity to infinity of: dx/[e^x + e^-x]

I cannot show my work because i have no idea how to do these problems. If someone can explain the direct/limit comparison tests then i will attempt these problems and post my work so you guys can guide me. Thanks.

Intuitively, if you are trying to show something converges, if you can find something larger that does converge, that would do it. Similarly, if you are trying to show something diverges and you can find something smaller that does diverge, that would settle that.

For example for your first problem. What would happen if you omit the sin(t) in the denominator. Would that make the integral larger or smaller? Would the resulting integral converge or diverge? Check it out and see if one of the comparison ideas work.