Redshift - How can $z$ be greater than 1? I'm having trouble understanding the equation for redshift:
$z = Δλ/λ ≈ Δf/f ≈ v/c$.
If $z = v/c$ and $c =$ speed of light,
how can $z>1$ (as nothing can exceed the speed of light)?
 A: The formula $z \simeq v/c$ is only approximately true when $v \ll c$. Redshifts greater than 1 are possible if the redshift is caused by relativistic motion or by cosmological expansion.
The cosmological redshift is not a Doppler shift and should not be interpreted as such except perhaps at very small redshifts. It is caused by the expansion of space between the time when the light is emitted and when it is received by an observer. This expansion could be interpreted as a recession speed at small redshifts, but as you have surmised, that interpretation runs into trouble when redshifts become greater than 1. It is the expansion of space that allows things to apparently recede at greater than the speed of light. Your statement that "nothing can exceed the speed of light" is more nuanced in General Relativity and has received many questions and answers in these pages.
A redshift larger than 1 is also possible when relativistic motion is applied to a Doppler shift. The correct formula is
$$ z = \sqrt{\frac{c+v}{c-v}} -1,$$
which can become arbitrarily large as $v \rightarrow c$.
If $v \ll c$, then the above expression can be approximated by
$$ z = (1  + v/2c +...)(1 + v/2c -...) -1 \simeq v/c$$
A: $\frac{\Delta\lambda}{\lambda}\approx \frac{\Delta f}{f}$ is only true when $\Delta\lambda \ll \lambda$.
So it isn't true when $z\ge 1$.
A: $\Delta \lambda$ can be very large, much larger than $\lambda$, because the wavelength can be quite stretched out:  a 21cm line earlier in the universe can be many meters now. 
A: If we denote z as a Doppler redshift (due to velocity of an object moving through space away from us) and z’ as cosmological redshift (due to the expansion of space), then conceptually, ONLY z’ should be applied to spatial expansion throughout the range of z’ and general relativity (GR) solved for distance as a function of z’. GR says nothing about velocities attached to spatial expansion, and the special relativity (SR) speed limit applies only to Doppler redshifts and objects moving through space. GR puts no limits  on the increase in distance from galaxy A (suppose our Milky Way) to unbounded galaxy B in time t as the universe expands. Only due to experimental limitations in the past was z instead of z’applied when z’< 1. But now, it is possible to distinguish outcomes between using SR and z versus GR and z’ even when z’= 0.01. If GR is applied to z’ < 1, the SR speed limit doesn’t apply. Just imagine a bound galaxy cluster getting further and further apart from all other such clusters like dots on a balloon that is increasing in size. SR forbids any object or information from passing another object faster than c, but no galaxy that gets further and further away from all unbounded ones due to expanding space can ever pass another galaxy. The SR speed limit simply doesn’t apply to spatial expansion.
