Here is a limit I received on a test the other day and I was wondering if there is another easier way to do it...and by the way you all have reformed me...I still love L'hopital's but I decided to be less lazy

Imputting the power series we get

Cancelling terms and changing the signs on the top right parenthases we get

I dont know how to display this in correct notation (suggestions would be helpful!) But what is concise mathematical notation to say that since we are going to 0 the only terms that affect the limit are the lowest exponent terms? I know you can use "Big Oh" notation in algorithim discreet mathematics but I have only seen that as things get sufficiently big not small...but anyways because of this fact this limit is equivalent to

Is this corect and/or is there an easier way? Thanks in advance!