BC 1-2
Limits 1
Name:
Nice and Not-so-Nice Limits
Nice Limits


Some limits seem obvious. For example, lim x 2  3 x  1  9 , as we would all guess.
x 2
This is because the function (x) = x2 + 3x – 1 is a well-behaved function near x = 2. In other
words, as x gets closer to 2, (x) gets closer to 9, so we say the limit equals 9. The behavior of
the function is predictable.
As long as the function is nice and well-defined in an open interval containing x = a, we
may evaluate the limit by simply evaluating the function at x = a. In this case the function is said to
be continuous at x = a.
(1)
lim  5 x  1 
(2)
x2
lim x  1 
(3)
x3
cos x

x 6 sin  3x 
lim
Not-so-Nice Limits
Unfortunately, lots of limits aren't quite so obvious and lots of functions are not wellbehaved at all points. Specifically, the limit we evaluate to find any derivative yields an
expression undefined when h = 0. In addition, we may wish to study the behavior of a function
near a value of x where the denominator is 0. In short, we need to understand more about limits.
 x2  x  6 
x2  x  6
lim 
and the point in
 . The function here is   x  
x 2  2 x  4 
2x  4


question is x = 2. Look at this problem numerically first. You may use the table on your
calculator to fill in the following chart. Under TBLSET, you will need to change the
Independent setting from AUTO to ASK in order to enter these x values directly into the table.
Example:
x
(x)
1.9
1.99
1.999
2
undef
What does the values in the table above suggest?
Now sketch the graph of .
(Try ZoomDec with xres = 1 on the TI-89.)
Describe and explain your graph.
BC 1-2
Lim 1.1
2.001
2.01
2.1
Example continued: Now, it's time to do some algebra. Factor, simplify the expression and
complete the following:
 x2  x  6 
lim 
  lim
x 2  2 x  4  x 2


Does your analytical work confirm your graphical and numerical work? Before you answer this,
x2  x  6
1
look at the two functions   x  
and g ( x)  ( x  3) . Is f ( x)  g ( x) ? ______.
2x  4
2
Explain carefully.
Does your answer bear on the question: “Does your analytical work confirm your graphical and
numerical work?”
Now
Many of these not-so-nice limits may be calculated by doing some algebra first,
simplifying, and then evaluating the resulting expression. Do the following limits analytically.
Confirm your work by using tables and/or graphics.
(4)
lim
x 2  3x  4
x1 x 2
 7x  6
3  x  h   3x 2
lim
h 0
h
2
(5)
BC 1-2
Lim 1.2
(6)
(7)
lim
x4
x 2
x4
2 2

lim x 5
x5 x  5
The following limits cannot be determined by simple algebraic methods. Use graphs and
tables to estimate/guess each limit.
sin  3 x 
lim
(8)
x 0
2x
ln x
x 1 x  1
(9)
lim
(10)
3x  9
x2 x  2
BC 1-2
lim
Lim 1.3