Where am I going wrong with this naïve derivation of the Fourier transform of cos(x)?

Question

When I naively attempt to compute the Fourier transform of cos(t) using the definition of the transform I end up with a limit as t goes to infinity involving sin(a*t). I can see how the Dirac-delta function and exponential form of cos(t) lead to a nice solution but I'm wondering where I'm going wrong.

Answer 1

If you integrate across a finite interval and then set the integration limits to infinity, those sincs that you obtained will converge to the Dirac delta in distribution. They do not converge in the ordinary sense.