How to evaluate limits for the Initial Value Theorem (Laplace)?

Question

I am working out the initial value for my function of s.

In the second line of working, every term in the numerator is divided by s^3, but every term in the denominator is divided by s (and not s^3).

Surely this changes the value of the fraction, as algebra normally doesn't allow you to do something like this.

Why does this work?

Answer 1

Each of the three brackets in the denominator is divided by $s$, so the whole denominator has been divided by $s^3$: $$ \frac{(3s+1)(s+3)(s+100)}{s^3} = \frac{(3s+1)}{s}\frac{(s+3)}{s}\frac{(s+100)}{s} = (3+s^{-1})(1+3s^{-1})(1+100s^{-1}). $$