Pre- L' Hopital's Rule
When evaluating limits of rational functions where both the numerator and the denominator equal 0 at the point x=a, without the help of L' Hopital's Rule, much algebraic manipulation is needed. We have to get either the numerator or the denominator to equal to something other than 0.
One way to do this is to divide both by x or a power of x, multiply both by x or by a power of x. Or divide both by the factor in the denominator or numerator that makes it zero. For example, for the limit of (x+3)(x-3)/(x-3)(x+5), try dividing both by x-3. Remember-- we are dangling our feet in the water-- x is not technically 3. It is just arbitrarily close to it! So we can do the indicated simplification!
Now, what do we do when something in the numerator or denominator decides to go to infinity as x=a? No Problem. Use division or multiplication to cancel that term out, or bring that term to the denominator. Remember! (any number)/infinity=0!
So, we can calculate the limit of the function (2x+2)/(2x^2+1) at x=infinity as follows:
1. Divide numerator and denominator by x.
2. Now the function's numerator does not contain any x's. This means it is a number other than infinity. The denominator, on the other hand, contains a big term-- 2x.
3. We safely conclude the limit is 0.
GENERAL RULE: Most of the times, if both the numerator and the denominator of a rational function is a polynomial, then when x approaches infinity--
1. if the numerator contains a higher power of x, the limit is infinity.
2. if the denominator contains a higher power of x, the limit is zero.
After- L' Hopital's Rule
Things become much more simple. Just note two things:
1. It is senseless to compute the limit of the derivative using L' Hopital's Rule, as the rule uses derivatives itself.
2. Don't keep differentiating! STOP when either the numerator or the denominator ceases to be zero!