Show that integrate.[e^(-4x)*(cos 3x)] from 0-infinity =4/25

Question

I got the answer(without applying limits) as, {(e^(-4x) )[3sin(3x) - 4cos(3x)]}/25

But when applying the upper limit what is the value of,
{(e^(-infinity) )[3sin(3*infinity) -4cos(3*infinity) ] }/25 ?

And while having a look at the answer I noticed that if the part we applied the upper limit is equal to 0,the rest,when we apply the lower limit,will give us the answer they've given..

Do e^(infinity) or sin(infinity) or cos(infinity) equal to zero by definition?

Answer 1

e^-∞ = 0
sin∞ and cos∞ are undefined, but lie between -1 and 1

multiplying by 0 leaves the result = zero

Answer 2

Thank you!