Discovery-Graphical and Numerical Limits
Purpose: To develop a graphical and numerical understanding of limits.
(NOTE: Do as much of this on your own as you can. We will discuss this next class period)
Idea of a limit: as the x-values get ”near” a certain number, how do the y-values behave?
limx→a f (x) = L Reads:
We are not interested in what the function does at x = a.
x2 − x − 2
Example: f (x) =
x−2
Press Y = and enter the above function into Y1 (be sure you put parentheses around both the
numerator and denominator). Graph using the W INDOW X=0 to 4.7 and Y=0 to 10.
Graph of f (x):
It looks like as x approaches 2, y approaches
What is the Y-value? Does this make sense?
. Press T RACE and move to X = 2.
Now press 2nd, T blSet and select Ask for Indpnt. Press 2nd T ABLE and use the table to answer
the following:
f (1.9) =
f (1.99) =
f (1.999) =
Then the limit as x approaches 2 from the left is given by limx→2− f (x) =
This is called a left-hand limit.
Use the table to compute the following:
f (2.1) =
f (2.01) =
f (2.001) =
Then the limit as x approaches 2 from the right is given by limx→2+ f (x) =
This is called a right-hand limit.
Since the left and right hand limits are equal, the limit exists and limx→2 f (x) =
Repeat the above experiment with f (x) =
happens?
1
.
x−2
(Use the Window X=0 to 4.7, Y= -10 to 10) What
Then we say limx→2 f (x) =
Repeat the above experiment with f (x) =
happens?
|x−2|
.(Use
x−2
the Window X=0 to 4.7, Y= -2 to 2) What
Then we say limx→2 f (x)
Summary There are three possibilities when taking the limit of a function:
1.
2.
3.