I need a pointer to a particualr method of formally proving limits of sequences. Our teacher gives a lot of problems for finding limits of expressions where both the numerator and denominator are really scary expressions of n (polynomials of various powers, factorials, trigonometric functions). He uses a method of quickly estimating the upper and lover bound of these expressions for n greater than... and then inserting a much simpler expression between the original and epsilon.
That is expr1 < expr 2 < eps. We look for numerator greater than original for n > .. and denominator smaller that original for n >.....

for instance $\displaystyle (5n+3n^3/2 +1)/(n^2-3n^3/2 - 7) < (12n^3/2) /(1/2n^2) < epsilon for n > 145 $ (here we have n^(3/2))
. The simper exp we can be easily solved for n.

The question is: is there a trick for quickly estimating the upper bounds or it just takes solving lots of problems and slowly getting a hang of it?