When I am computing a limit where the degree of the numerator is different from the degree of the denominator or vice versa what would I multiply both by in order to solve. Also if there is a square root does it change the what I would multiply by. It's easy to solve the simple limits however I am looking for a set of rules to follow when I am trying to solve the more complex problems.
There are no "rules" as such, just methods that have been shown to work. And there is usually more than one way to think about them. So you have to work some problems and find what you are most comfortable with.
Note that as x goes to infinity only the terms of largest degree will be important. So we can do this in two ways:
1) Keep only the largest degree terms in the numerator and denominator:
2) We can reduce the order of the numerator and denominator by dividing both by the largest power term, in this case :
Notice that any term , so this becomes:
These processes can be done using any degree polynomials in the numerator or denominator. Also it does NOT matter that we are taking a limit as x goes to infinity, you can use similar arguments as x approaches any number as well.