= Solution
A finite <optical pupil> spreads a point into a <point spread function>. In <incoherent imaging> an extended <intensity> distribution is convolved with that function; its <Fourier transform> is the <optical transfer function>, which attenuates and ultimately removes high spatial <frequencies>. For an ideal circular pupil the point response is an <Airy disk>. The <Rayleigh criterion> for two equally bright incoherent points is
$$
\boxed{\theta_R\simeq1.22\lambda/D},\qquad \Delta x_R\simeq1.22\lambda f/D.
$$
A microscope's object-space scale is of order $\lambda/(2\,\mathrm{NA})$, with the numerical factor depending on illumination and the criterion; $\mathrm{NA}=n\sin\alpha$ measures the accepted ray cone. A larger pupil or shorter <wavelength> improves the diffraction scale. Aberrations, <defocus>, detector pixel sampling, motion, background and <photon> noise can worsen practical performance. Merely magnifying the same blurred image adds no information. Coherent sources require amplitude <superposition> instead of the incoherent <intensity> rule. Rayleigh resolution is a convention for separating a pair, distinct from the precision of localizing one isolated point; information does not abruptly vanish below it.
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