ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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Activity A1 – Finding Limits Graphically
Often we want to analyze a function’s values near a specific domain value. For example,
we may want to investigate the y values of the graph(RS1) shown at right near x = –2. To
do this let’s use the limit applet(RS2). On the limit applet, select the option for a 2 part
piecewise function as shown below.
Now input the piecewise function
 2 x  1 3  x  2
f ( x)  
 x  2 2  x  2
into the applet as shown below. Be sure to set the inequality cells correctly.
Click on update graph to have the applet graph the piecewise function. Before we can
investigate the values of the function near x = –2 we need to set the cell shown below to
–2.
First let’s evaluate the function at x = –2. To do this, click on the Value button. The
value of the function at x = –2 is recorded in the blank space below the buttons where it
reads “f(–2) = –4.”
Now lets investigate the behavior of the y values as we approach x = –2 on its left side.
To do this, click the Left Limit button. Notice the blue dot approaches x = –2 from the
left. We call this the left-hand limit. The value of the left-hand limit is recorded in the
ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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blank space below the buttons where it reads “the left-hand limit at x = –2 is –3.”
Mathematically we write
lim f ( x)  3 .
x 2
Now let’s investigate the behavior of the y values as we approach x = –2 on its right side.
To do this, click the Right Limit button. Notice the red dot approaches x = –2 from the
right. We call this the right-hand limit. The value of the right-hand limit is recorded in
the blank space below the buttons where it reads “the right-hand limit at x = –2 is –4.”
Mathematically we write
lim f ( x)  4 .
x 2
Since the left-hand limit does not equal the right-hand limit, we say that the limit of f(x)
at x = –2 does not exist. This can be shown on the applet by clicking on the Limit button.
Notice, when the button is clicked, the red and blue dots approach two different y-values.
Because of this the limit does not exist. This conclusion is recorded in the blank space
below the buttons where it reads “the limit at x = –2 does not exist.” Mathematically we
write
lim f ( x)  DNE .
x 2
The example above demonstrates that the left and right hand limits of a function must be
the same for the limit of the function to exist. This leads to the definition of the limit of a
function at a point x = a(RS3).
1. Use the limit applet(RS2) to find the following given
 2 x 2  x  1
f ( x)  
.
x  3 1  x  2
a) f (1)
b) lim f ( x)
x 1
c) lim f ( x)
x 1
d) lim f ( x )
x 1
2. Use the limit applet(RS2) to find the following given
  x2
5  x  2

f ( x)   3
x  2
 x 2  4 x 2  x  5

a) f (2)
b) lim f ( x)
x 2
c) lim f ( x)
x 2
d) lim f ( x)
x 2
ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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The results of problem 2 demonstrate an important concept regarding limits. You should
have noticed that lim f ( x) existed even though the value of the function was different
x 2
than the limit of the function (i.e. lim f ( x)  f (2) ). When finding limits of functions,
x 2
this may happen because the limit of a function represents the value of the function near
a specific x value not at an x value. This is an important concept to understand about
limits.
3. Use the limit applet(RS2) to find the following given
2 x  1 3  x  1

f ( x)   2 x  3 1  x  1
 x 1
1 x  2

a) f ( 1)
e) f (1)
b) lim f ( x)
f) lim f ( x)
x 1
x 1
c) lim f ( x)
g) lim f ( x)
d) lim f ( x)
h) lim f ( x )
x 1
x 1
x 1
x 1