Hi, I'm having a terribly difficult time deciphering whether an improper integral is divergent or convergent.

From what I notice, is it divergent whenever the answer is infinity? and convergent whenever the answer equals a specific number?

I just got an answer that is:

the limit as a approaches negative infinity -1/2a^2 + the limit as b approaches infinity 1/2b^2

is the answer infinity because if you plug any number into a it gets infinitely smaller and if you plug any number into b it gets infinitely bigger? is that explanation correct?

I also just got another answer that is:

the limit as b approaches negative infinity (1/2-1/2e^2b)=1/2 ............how does 1/2e^2b as b is approaching negative infinty equal zero for that to be true? I don't see that at all. Thanks in advance for any help.