This looks like a fairly straightforward problem but it's been bugging me for a while

$$y=\frac{1-\frac{1}{e^{bx}}}{x}$$

How do I show that when x = 0 , y = b ?

It is a transcendental expression, transcendental expressions aren't reducible, did you pass basic algebra? An example of reducible expression is $$\frac{x^2-y^2}{x-y}$$. Does the expression in your exercise look like a ratio of two polynomials?

So is it true that for an expression to be reducible, it must be a ratio of polynomials ( where x, and y can be a function) or is there more subtle cases?

If there is more subtle cases, then in general, how to prove that an arbitrary expression f(x)/g(x) is irreducible?