I am a math teacher and I have to teach a topic called "Bruchterme" and "Bruchgleichungen" in german (I don't know the english word for it). For example
$$ \frac{x^2 - 3}{(x - 2)x^2} + \frac{4}{x} + 2 $$
is a "Bruchterm" and
$$ \frac{4x}{2x -3} = 4 - \frac{2x}{x-1} $$
is a "Bruchgleichung".
Students have to learn, how to determine for which $x$ the term or equation is defined (i.e. the "singularities") and how to solve such equations.
Now my problem is that most textbooks about this offer little to no interesting applications of this, especially no applications of determining for which $x$ the term is defined.
Now I am looking for interesting examples from classical physics or engineering for this type of problem. Especially examples where singularities occur and are physically "interesting" in some way. Though it is for high school level, I don't want to restrict the question to trivial examples, but to classical physics.
The only physical applications I have in mind are the following:
Two parallel resistors $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$
Same type of equation: $\frac{1}{f} = \frac{1}{b} + \frac{1}{g}$ for the thin lens equation (where $b$ and $g$ are the distances from the image to the lens resp. from the object to the lense)
However here the denominators are rather trivial and I don't see why it would be interesting to determine for which value for example of $b$ (if the other values are fixed) the equation is not defined.
Gravitational force ($\propto \frac{1}{r^2}$) here one might discuss that the force goes to infinity for $r \to 0$. However the point of view of the curriculum is not to discuss such limits but just to look if one is allowed to plug in certain values for the variable or not (in a sense that dividing by 0 is not allowed or not defined without connection to limits).
Are there some interesting examples from physics where students see that it is worth to learn how to solve such equations (from type and perhaps complexity as in my example above)
Are there examples from physics where students see that it is worth to learn to determine for which values the term or equation is defined?
Edit (too long for a comment)
@Danu: I am teaching in the less known part of the german education system, called "Berufliche Schulen" more specifically I am at a school which focus is on technology, engineering and science. The system is a bit complicated with classes on very different levels. To make it short I have the motivation problem described above in a low level class and in a high level class (the level of the second one is equivalent to a German "Gymnasium" or even a bit higher in mathematics and physics) but the students are generally very interested in technology, engineering, physics and computer science (most of them want to take up a profession in this fields. Ranging from electricians or lab technician in the lower level classes up to enegineers or physicists in the higher level classes). Students are from 15 to 18 years old. Even in the lower level classes they are very interested in nontrivial physics even if they cannot grasp it conceptually or mathematically. So for motivation purposes some nontrivial, but for the students interesting examples would be good.