A. Limits can be calculated very quickly and are closely related to horizontal asymptotes.
B. There are many ways limits can be used:
Ex
[1] 
a. The answer is f(2) or 22 - 4 = 0.
a. If g(n) = 0 and f(n) does not, then the limit is undefined.
b. If g(n) = 0 and f(n) = 0, then use L'Hopital's Rule. See Below.
c. If g(n) does not equal 0, then the answer is f(n)/g(n)
a. Since g(4) = 2 and not 0, the answer is f(4)/g(4) = 3(16) + 1/2= 49/2.
a. Problems like these have the same basic rules as horizontal asymptotes.
b. If the
degree of the numerator is greater than the degree of the denominator the answer
is 
.
c. If the degree of the numerator is less than the degree of the denominator the answer is 0.
d. If the degrees are the same the answer is a/d where a is the coefficient in front of the highest degree in the numerator and d is the coefficient in front of the highest degree in the denominator.
a. Since the degree of the denominator is 4 and the degree of the numerator is 3 and 4>3, the answer is 0.
a. Since the degrees are the same the answer is 4/2 = 2.
1.
L'Hopital's Rule is a very handy rule that makes solving problems very
easy. It states that when taking the limit of f(x)/g(x) you
get the value of 0/0 or 
/
,
then the limit is equal to f'(x)/g'(x). (See derivatives)
a. If I plug in 2 into the equation I get 0/0, which means I can use this rule.
b. The derivative of the numerator is 2x - 4.
c. The derivative of the denominator is 2x.
d. Therefore the limit is equal to 2(2)-4/2(2) = 0/4 = 0.
e. The answer is 0.
a. Notice, that when I plug in 3, I do in fact get 0/0.
b. The derivative of the numerator is 2x - 4.
c. The derivative of the denominator is 2x - 2.
d. Therefore, the limit is equal to 2(3)-4/2(3)-2 = 2/4 = 1/2.
e. The answer is 1/2.
a. In this problem, we get 0/0, so we can use this rule.
b. The derivative of sin 3x = 3 cos 3x.
c. The derivative of x = 1.
d. So 3*cos 3(0)/1 = 3 x 1 = 3.
e. The answer is 3.
D. Instead of using L' Hopital's Rule, you can factor the equations and cancel something out, but this rule is much faster and easier.