Periodic functions $f(t)$ (with a time period $T$)
can be approximated by a Fourier *series*,
i.e. by summing harmonic oscillations with the *discrete* frequencies
$0, \frac{2\pi}{T}, 2\frac{2\pi}{T}, 3\frac{2\pi}{T}, \ldots$ .
$$f(t)=\sum_{n=-\infty}^{+\infty} F_n e^{in\frac{2\pi}{T}t}$$

But, as you already noticed, with a Fourier series you
cannot build *aperiodic* functions
(e.g. functions limited to a finite range).
For approximating such functions you need a [Fourier *integral*][2],
i.e. summing harmonic oscillations with *all*
frequencies from $0$ to $\infty$.
$$f(t)=\int_{-\infty}^{+\infty}F(\omega)e^{i\omega t}\ d\omega$$

 [1]: https://en.wikipedia.org/wiki/Fourier_series
 [2]: http://www.math.toronto.edu/courses/apm346h1/20151/L15.html