Finding Limits Using Limit Laws:
Finding limits algebraically allows us to find the limit of a function exactly.
The following laws will allow us to calculate limits algebraically.
Limit Laws:
Suppose that c is a constant and that the following limits exist:
and
lim f ( x)
x a
lim g ( x)
x a
Then:
lim [ f ( x)  g ( x)]  lim f ( x)  lim g ( x)
1)
xa
xa
xa
Limit of a Sum
2)
lim [ f ( x)  g ( x)]  lim f ( x)  lim g ( x)
Limit of a Difference
3)
lim [cf ( x)]  c lim f ( x)
Limit of a Constant Multiple
4)
lim [ f ( x) g ( x)]  lim f ( x)  lim g ( x)
5)
lim
x a
xa
x a
xa
x a
xa
xa
xa
f ( x)
f ( x) lim
 xa
g ( x) lim g ( x)
xa
if lim g ( x)  0
xa
Limit of a Product
Limit of a Quotient
xa
6)
lim [ f ( x)] n  [lim f ( x)] n
xa
xa
where n is a
Limit of a Power
positive integer
7)
lim
xa
n
f ( x) 
n
lim f ( x)
xa
where n is a
positive integer
Limit of a Root
Using the Limit Laws
1)
Use the limit laws and the graphs of f and g in the below diagrams to
evaluate the following limits.
a)
lim [ f ( x)  5 g ( x)]
c)
lim
x 1
x4
f ( x)
g ( x)
b)
d)
lim [ f ( x) g ( x)]
x  2
lim [ g ( x)]3
x3
Using the Limit Laws
2)
Evaluate the following limits and justify each step:
a)
lim (6 x 2  3x  1)
x 3
b)
x3  2x 2  1
x  2
5  3x
lim
Limits by Direct Substitution:
If f is a polynomial or rational function and a is in the domain of f , then
lim f ( x)  f (a )
xa
f (a ) has to exist to use direct substitution
Functions with this direct substitution property are called continuous at a . (you
will learn more about this in calculus)
Finding Limits by Direct Substitution
3)
Evaluate the following limits:
x 2  6x
x4  3
a)
lim
c)
x 2  5x
x 1 x 4  2
x2
lim
b)
lim (2 x 3  10 x  8)
x3