Why we can still use this function when it's dividing zero my teacher gave my class a question in calculus but the method he used to attempt the question was kind of odd.
Question: An aircraft at a constant 500 metres height is flying towards an observer at 80 m/s. How is the angle of elevation changing if the aircraft is overhead?
He did:
let the elevation angle be $\theta$. then we have function:
tan $\theta$ = $500/x$ where x is the honrizontal distance between the plane and the observer.
  Further it's just diffrentiate with respect to t and substitution (since we know $dx/dt$ is -80)
Finally we have:
$\frac{d\theta}{dt}$=$\frac{4000}{x^2}$ $\times$ $\frac{x^2}{h^2}$
My teacher told us just to cancel out the $x^2$. But isn't the function undefined when x is zero?
Thank you.

It's kind of like cancelling the $x$'s in the equation $y = \frac{x(x+3)(2x+5)}{x}$ when you try to find the limit as x approaches 0. The function is undefined at x = 0, but the limit still exists and still makes sense. The function $y = x(x+3)(2x+5)$ matches everywhere else except x = 0, similar to your problem with the cancelling $x^2$. 
