Interference for for N slits formula

[Nikitin]

Interference for a lattice with N slits

OK, so check out the first formula (the one for intensity) in the following pdf:

http://folk.ntnu.no/magnud/OvLf/bolge/oving11.pdf

It's the formula for Intensity as a function of angle from the interference for a lattice with N very thin slits.

Mathematically, why is $I_{max}$ found only when BOTH the numerator and denominator are approaching zero?

I agree that the intensity is largest when the ratio between the numerator and denominator is biggest, but why does this only happen when both approach zero?

 

[Philip Wood]

Replaces previous post, which was misleading.

Consider when $\theta$ is close to $n\pi$. Put $\theta = n\pi + \epsilon$.

Then $\frac{sin N\theta}{sin \theta} = \frac{sin (Nn\pi +N\epsilon)}{sin (n\pi + \epsilon)} = \frac{cos (Nn\pi) sin(N\epsilon)}{cos (n\pi) sin\epsilon} = ± \frac{sin N\epsilon}{sin \epsilon}$

The limit of this as $\epsilon$ approaches zero is simply $±\frac{N\epsilon}{\epsilon} = ±N$.

 

[Nikitin]

Oh I just figured it out. When the upper term approaches zero, then so must the lower turn because N is just a whole number.

thanks