On this website I found the following drawing of the magnitude spectrum of a sine wave along with the derivation of the Fourier Transform of $\sin(x)$, given by $X(f) = \frac{1}{2j}\left[\delta(f−f_0) - \delta(f+f_0)\right]$, and the following explanation:
The term $\delta(f−f_0)$ is a delta function in the frequency domain which is shifted right by frequency “$f_0$” .Its amplitude is $+\frac{1}{2}$ and the term $\delta(f+f_0)$ represents a delta function which is shifted left by frequency “$f_0$”.Its amplitude is $-\frac{1}{2}$.
To me, this seems to suggest that $\delta(0) = 1$, but this does clearly not coincide with the definition of the Dirac delta function. For, as Wikipedia states, "the Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, $$ \delta(x) \simeq \begin{cases} +\infty & \text{if } x=0\\ 0 & \text{if } x\neq 0 \end{cases} $$ and which is also constrained to satisfy the identity $\int_{-\infty}^{\infty} \delta(x) \text{d}x = 1$."
I would really appreciate any help to clear up my confusion about this!