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Q) Prove that (2n^4 + 4n^2 + 3n - 5)/(n^4 - n^3 + 2n^2 - 80) converges to 2 as n goes to infinity.

A)

By the algebra of limits, this converges to 2 since

lim(n->oo)[2 + 4/n^2 + 3/n^3 - 5/n^4]/lim(n->oo)[1 - 1/n + 2/n^2 - 80/n^4)

(2 + 0 + 0 + 0)/(1 - 0 + 0 + 0) = 2

However, I would like to do this a little more precisely and rigorously. Can someone tell me...

Would I fix epsilon (e) > 0.

Then the absolute value of the quotient minus 2 must be bigger than epsilon, etc...

Or would I approach this in another way.

Any Help woudl be appreciated.

Thanks.