math 265A introduction to limits of functions TI83/84
In this lab we shall investigate the behavior of a function f near a specified point. While this is
sometimes a straightforward process, it can also be quite subtle. In many cases, the process for finding a
limit must be applied carefully. By gaining an intuitive feel for the notion of limits, you will be laying a
solid foundation for success in calculus.
1) Consider the function
()
1
1
4
−
−
=x
x
xf .
a) By successive evaluation of f at x = 1.9, 1.99, 1.999, 1.9999, what do you think happens to the
values of f as x increases towards 2?
b) Do a similar evaluation of f for values of x slightly greater than 2.
For example, let x = 2.1, 2.01, 2.001, 2.0001.
As a shorthand and anticipating a forthcoming definition, we shall describe what you found in parts (a)
and (b) by writing
or more specifically
()
15lim
2=
→xf
x15
1
1
lim 4
2=
−
−
→x
x
x
c) Note that in this particular case you could have “CHEATED” by immediately evaluating f at 2.
Use a graphing utility to graph f . Use a window setting with XMin = -4, XMax = 4, YMin = -4,
and YMax = 25.
2) Use the same function f as above, but this time consider what happens to the values of f as x
approaches 1.
a) Study this situation experimentally as you did in parts (a) and (b) of problem 1, but this time
choosing x values close to 1.
What are your conclusions and, in particular, what is
(
)
xf
x1
lim
→?
b) What happens when you try to “CHEAT” as was done in part (c) of problem 1?
Use a window setting with XMin = -4.7, XMax = 4.7, YMin = -1, and YMax = 11.4.
Do you see why “CHEATING” DOESN’T PAY in this situation?
There are situations in which direct evaluation at the specified point is possible and actually give
the limit. These give rise to a concept called continuity. However, there are many important
situations that arise in calculus when this technique will not work.
