How to Apply L'Hopital's Rule to Probability Equations?

[Crazy Gnome]

The problem statement

Using the Equation

P($$\theta$$)= P₁[ $$\frac{sin(Nkdsin(\theta)/2)}{sin(kdsin(\theta)/2)}$$ ]²

show that the probability at sin($$\theta$$)=j$$\frac{\lambda}{d}$$, where j is an integer, is P($$\theta$$=sin^-1(j$$\lambda$$/d))=N²P₁

Hit: find $$\frac{sin(Nkdsin(\theta)/2)}{sin(kdsin(\theta)/2}$$ as sin($$\theta$$) approaches j($$\lambda/d$$) using L' Hopital's rule.



My problem: I am not sure how to apply L Hopital's rule to this situation. What would be my F(x) and what would be my G(x)?

 

[badphysicist]

This looks like a problem from diffraction theory, but here's a little help. Basically L'Hopital's rule is used when the limit as an equation that can be expressed as a fraction of two equations diverges. L'Hopital's rule says to find the limit of the derivative of the numerator over the derivative of the denominator.

 

[tiny-tim]

I am not sure how to apply L Hopital's rule to this situation. What would be my F(x) and what would be my G(x)?

Hi Crazy Gnome!

Your x can be either θ or sinθ …

it makes no difference, the result will be the same.

(Personally, I'd use θ. )