SECTION 1.2:
FINDING LIMITS GRAPHICALLY AND
NUMERICALLY
Goals: The Student Will Be Able To:

Create a useful table of values, and use it to
evaluate a limit (numerical analysis)

Create a hand-sketched graph, and use it to
evaluate a limit (graphical analysis)

Use a graphing utility to set up a table and trace
graphs, so to evaluate a limit

Determine whether the graphical or numerical
approach is more appropriate

Determine and explain when and how certain
types of limits do not exist (jump, infinite,
oscillating)

Acquire and use new associated terminology and
symbolic notation
1
INTUITIVE SENSE OF A LIMIT
(2.5, 4)
(2.9, 4.8)
(2.99, 4.98)
(2.9999, 4.9998)
(_______,_______)
(3.0001, 5.0002)
(3.01, 5.02)
(3.1, 5.2)
(3.5, 6)
2
GENERAL DESCRIPTION OF A LIMIT
The word “LIMIT” is abbreviated: ”lim”
It is used in reference to the value of a function or
expression. It denotes the “y” value that a function
or expression is getting closer and closer to, as
corresponding “x” values approach a fixed number.
Symbolically:
lim f ( x)  L
x c
The “c” is some particular/specific “x” value. One must
consider values a little bit bigger and a little bit smaller
than this “c” value in evaluating a limit. A limit
disregards what is happening at “c”.
The “L” is the resulting “y” value, known as the limit.
It is what appears to be the value that f(x) is “settling in
on”, as you evaluate it in a close “neighborhood” of
values around “c”.
3
EXAMPLES OF LIMIT EVALUATIONS
lim x 2  9
lim x  0
lim sin x  1
x 2
1
lim 2  
x0 x
1
lim  d .n.e.
x0 x
x 8
lim
1
x8 x  8
x3
x 0
2
x

25
lim tan x  d .n.e. lim
 10
x 2
x5 x  5
1
lim  0
x x
1
lim  
x 0  x
4
NUMERICAL EVALUATION OF A LIMIT
Consider the following function:
x3  1
f ( x) 
x 1
Can you evaluate the function for various “x” values? …
f(2); f(3); f(0); f(-1); f(-2) … f(1) ???
3
3
1
2 1
f (3) 
 13
f (2) 
7
3 1
2 1
f (0)  1
f (1)  1
f (2)  3
3
Is there one value of “x” which will cause a problem? …
But of course there is!!! There is one value not in the
DOMAIN of f(x) 
x 1
13  1 0
f (1) 
  ???
1 1 0
There is NOT a corresponding “y” value for “x”, when x = 1.
But we can examine the “y” values, for “x” values that are
“near” (in a neighborhood) around x = 1. This is the idea of a
LIMIT.
5
x3  1
f ( x) 
x 1
Instead of asking:
f (1)  ? … Ask:
lim f ( x)  ?
x1
One way to answer this is by using a numerical approach
(there will be others).
From the right
From the left
X
f(x)
.9
2.710
.999
2.997
1
???
1.001
3.003
1.1
3.310
You can move arbitrarily closer and closer to x = 1, and
witness how y will move ever closer to y = 3.
x3  1
lim
3
Thus we can say: x 1 x  1
“The limit of the function as x approaches 1 is 3”
Here … try one on your own…
Do it numerically!!!
x 2  3x  2
lim
?
x 2
x2
6
GRAPHICAL EVALUATION OF A LIMIT
x3  1
f ( x) 
Consider the same function again:
x 1
However, this time, examine the LIMIT question by looking
at the graph. Let’s use our TI83 to do this.
x3  1
Y1  f ( x) 
x 1
Enter into Y=
3
1
then
ZOOM…4:ZDecimal…
WINDOW…Xmin/2…
Xmax/2…Ymin=0…
Ymax*2…GRAPH
OH MY!!! Is your calculator broken? No, the “hole” in the
graph is the visual effect of the break in the domain at x = 1.
Use TRACE to explore around the “hole”. See what happens
to “y”, when the cursor is at x = 1.
Use TRACE…ZOOM…2:Zoom In…ENTER a few more
times to visually inspect the “hole” neighborhood even more
closely. Does the graph confirm that the limit is 3?
7
EXAMPLE 1:
Find the limit numerically, and then graphically:
lim
x0
X
f(x)
x
x 1 1
-0.01
1.9950
Do you see why f(0) is a problem?
-0.0001
1.9999
0
???
0.0001
2.0001
0.01
2.0050
It should be quite obvious that as x  0; y or f(x) 2
 lim
x 0
x
x  1 1
2
… Confirm this by graphing.
ZOOM…4:ZDecimal…
WINDOW…Ymin=0…
GRAPH
The “hole” in the graph
will only be visible if you turn
off your axes. It should
become clear that the “hole”
is at (0,2).
8
EXAMPLE 2:
Find the limit of the “piecewise function”:
1 x  2
f ( x)  
0 x  2
1
lim f ( x)  ?
x 2
1
2
We can look at it both graphically and numerically. The
graphical evaluation makes better sense. What is the “y”
value as x approaches 2 from the left and the right? The “y”
value stays at a constant of 1, regardless of how close you are
to x = 1. The big point is that, what is happening AT x = 1 is
irrelevant!!! Imagine 2 bugs walking on the curve,
approaching the “hole” from the left and right. They will be
able to shake hands. As they “meet over the hole”, what are
their “y” values?
lim f ( x)  1
x 2
even though f (2)  0
Numerically:
X
f(x)
1.9
1
1.9999
1
2
0
2.0001
1
2.1
1
9
EXAMPLE 3:
Show that the following limit does not exist.
lim
x 0
1
x
x
-1
Consider the graph, and what is happening for positive x
values that are getting closer to 0.
Now consider the graph for the negative x values that are
getting closer to 0.
Can you imagine how frustrated the two bugs are, who
cannot shake hands?
Since the “right-hand” y value is at 1, and the “left-hand” y
value is –1, the overall limit DOES NOT EXIST! … because
the left and right behavior differs.
Numerically:
X
f(x)
-0.01
-1
-0.0001
-1
0
???
Disagreement
.0001
1
0.01
1
10
EXAMPLE 4:
Discuss the existence of the limit:
1
lim 2
x 0 x
Consider the graph, and what is happening for positive x
values that are getting closer to 0. “y” is getting bigger and
bigger, with no end in sight.
Now consider the graph for the negative x values that are
getting closer to 0. The same thing is happening to “y”
The two bugs are frustrated again, but for a different reason.
They never settle in on a final “y” value”. And, it’s as if they
have a thin glass wall between their feet.
Numerically:
X
f(x)
-0.01
10000
-0.0001
100000000
0
???
.0001
100000000
0.01
10000
1
lim 2  d .n.e. …or another way to better say this is:
x 0 x
1
lim 2   … the function “increases without bound”.
x 0 x
11
EXAMPLE 5:
Discuss the existence of the limit:
1
1
lim sin
x 0
x
-1
Consider the graph, and what is happening for positive or
negative x values that are getting closer to 0. “y” appears to
be oscillating back and forth between extremes of 1 and –1.
Again, since there is not one specific value that the function
is settling in to, there is no limit.
The numerical approach can confirm this, as well. Here is a
unique numerical view from the right to illustrate the
unending oscillation.
Numerically:
x
sin
1
x
2

2
3
2
5
2
7
2
9
2
11
x0
1
-1
1
-1
1
-1
Limit d.n.e.
12