How Fourier Expansion indicates the Amplitude w.r.t a certain frequency?

[genxium]

Homework Statement

I want to know how could I extract the amplitude(of the sinusoid component) of a random continuous wave w.r.t a certain frequency response? The teacher said the Fourier Expansion can do that but I'm really confused by the limits and integrals.

Homework Equations

$F(\omega)=\int f(t) e^{-j \omega t} dt$

The Attempt at a Solution

I tried the very easy example and want to extract the amplitude where the frequency matches(say $\omega = \omega_0$). $f(t)=A \cdot cos\omega_0t,\hat{f}(t)=A \cdot e^{j \omega_0 t},\hat{F}(\omega)=\int \hat{f}(t) \cdot e^{-j \omega t} dt$, range ($-\infty ,\infty$), but it turned out to be $Re\{ \hat{F}( \omega ) \}=A \cdot \frac{sin(\omega_0 - \omega ) (t_2-t_1)}{\omega_0 - \omega}$, where $t_2=\infty,t_1=-\infty$ , it's weird if I follow the basic operation of sin function, I got $Re\{ \hat{F}( \omega ) \}=2 \cdot A \cdot \frac{sin(\omega_0 - \omega ) \infty}{\omega_0 - \omega}$, and then although applying that $lim \frac{sinx}{x} -> 1$ while x->0, it's 2A, besides I don't even know if this's right.

I have no idea what happened...

Any help will be appreciated !

 

[genxium]

Terribly sorry that my teacher has corrected my mistakes, the Amplitude should be $|F(\omega)|$ instead of $Re \{ \hat{F}(\omega) \}$,but the calculation becomes even harder, I'm still trying on this question.