If is any polynomial then

This is trivial and can be proved easily using properties from Limits of a Function

Firstly due to Property 5 in Limits of a Function > Arithmetic Rules for Limits of Functions we can say that if

If then due to Real Analysis Theorems > Theorem 3 we can say that for limit to exist

Since , must be a factor of both the Polynomials and we can write them as where is another polynomial function. Similarly for .

Let $f(x) = \frac{P(x)}{Q(x)}

Step 1: If , then

Otherwise, go to Step 2.

Step 2: If but , then the limit does not exist. Otherwise, go to Step 3.

Step 3: If and , write

and return to Step 1 using the new function

since .