Pre-Calculus
Limits (An Intuitive Approach)
Homework:
Objective: SWBAT understand the basic concept of “limit.” The concept of a “limit” is the
fundamental building block on which all calculus concepts are based. Students must understand
this concept to fully appreciate calculus.
Limits (An informal View)
If the values of f ( x) can be made as close as we like to L by taking values of x
sufficiently close to a (but not equal to a) then we write
lim f ( x)  L
x a
which is read “the limit of f ( x) as x approaches a is L” or “ f ( x) approaches L as x
approaches a.”
1. The following table gives values for the function f ( x ) 
x (Radians)
f ( x) 
 1.0
 0.9
 0.8
 0.7
 0.6
 0.5
 0.4
 0.3
 0.2
 0.1
 .01
sin x
.
x
sin x
x
.84147
.87036
.89670
.92031
.94107
.95885
.97355
.98507
.99335
.99833
.99998
a. It appears that the function is approaching 1 when the values for x are approaching zero from
the right. The notation for this would be
lim
x 0
sin x
1
x
which is read “the limit of
sin x
as x approaches a from the right is 1”
x
b. It appears that the function is approaching 1 when the values for x are approaching zero from
the left as well. The notation for this would be
lim
x  0
sin x
1
x
which is read “the limit of
sin x
as x approaches a from the left is 1”
x
c. Since the limit from the right and the limit from the left are equal we can write
lim
x 0
sin x
1
x
which is read “the limit of
sin x
as x approaches a is 1”
x
d. The notation in parts a and b above are called “ONE SIDED LIMITS” and the notation in
part c is called a “TWO SIDED LIMIT”. A two-sided limit can only be used if the limit from
both the left and the right are equal. The following notation summarizes the concept of a twosided limit.
lim f ( x)  L if and only if
x a
lim f ( x)  lim f ( x)  L
x a 
x a 
The value of f at x  a has no bearing on the limit as x approaches a. This is very
important to remember.
2. Use the graph to find the limits.
a.
old book #2
c.
e.
lim f ( x) 
b.
lim f ( x) 
d.
f (2) 
lim f ( x) 
f.
lim f ( x) 
x  2
x 2
x 
lim f ( x) 
x  2
x 
3. Use the graph to find the limits.
a.
Old book # 5
c.
e.
4. Limits at vertical asymptotes
lim f ( x) 
b.
lim f ( x) 
d.
f (2) 
lim f ( x) 
f.
lim f ( x) 
x  2
x 2
x 
lim f ( x) 
x  2
x 
5. Use the graph to find the limits.
a.
c.
Old book #7
lim  ( x) 
b.
lim  ( x) 
d.
 (2) 
lim  ( x) 
f.
lim  ( x) 
x 2
x 2
e.
x 
6. The Greatest Integer Function
f ( x)  x
12
10
8
6
4
2
-15
-10
-5
5
-2
-4
-6
-8
-10
-12
a.
b.
lim1 x 
x 2
lim x 
x  52
c.
lim x 
d.
lim x 
x2
x 0
10
15
lim  ( x) 
x 2
x 