Download Math 112 Lecture 1: Introduction to Limits

Why Limits?
Limits are used in calculus to
◮
define and calculate the slope of a tangent line
◮
calculate velocities and rates of change
◮
define and calculate areas and volumes
But, more generally, limits provide a way to extend the operation of function
evaluation.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
2/22
Example 1 – An Indeterminate Form
Consider the function
f (x ) =
x +1
x2 − 1
The domain of this function is
Clint Lee
Math 112 Lecture 1: Introduction to Limits
3/22
Continuing Example 1
Evaluating f at x = 1 gives
f (1) =
2
0
which is obviously undefined. Use your
graphing calculator or Maple to plot the graph
of
this function
in the
viewing rectangle
−2, 3 × −4, 4 .
Does your graph look like this? What’s wrong with this picture?
Clint Lee
Math 112 Lecture 1: Introduction to Limits
4/22
Continuing Example 1
Evaluating f at x = −1 gives
f (−1) =
0
0
which is again undefined. However, now we do not necessarily get an
arbitrarily large value, as in the last case. This is an indeterminate form.
Something different happens at x = −1 than at x = 1. The difference is
hinted at by the graph we saw earlier. There is an open circle on the graph
at x = −1.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
5/22
Continuing Example 1
To investigate what happens to f at
x = −1 make a table of values near
x = −1.
Clint Lee
x
f (x )
Math 112 Lecture 1: Introduction to Limits
6/22
Definition of Limit
From Example 1 we see that even though f (−1) is not defined there is a
finite and precisely defined limiting value for f at x = −1. We say that
1
x +1
=−
2
x →−1 x − 1
2
lim
The following definition makes this more precise.
Definition (Limit of a Function)
We say that
lim f (x ) = L
x →a
if f (x ) can be made arbitrarily close to L by making x sufficiently close to a,
on either side of a.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
7/22
Example 2 – Applying the Definition
To apply the definition, suppose that we
want to make sure that the value of f (x )
is within 0.2 of limit value L = −0.5. This
means that we want
The graph shows the relation between these two intervals.
− 1 .5
− 0 .5
− 0 .3
− 0 .7
−0.7 ≤ f (x ) ≤ −0.3
From the table of values in Example 1 we see that
and
Hence, we only need to have x within
of the limit point x = −1, so that
Hence for x in the interval
the values of f (x ) are in the
interval
, which is contained in the interval
. The graph of the function f lies entirely in the box bounded
by the horizontal lines and the vertical lines.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
8/22
Continuing Example 2
If we decrease the vertical extent of the box, which means we want to keep
the values of f (x ) closer to the limit value, we must decrease its horizontal
extent as well. From the graph it appears that no matter how small we make
the vertical extent of the box, we can make the box narrow enough
(horizontally) so that the graph of f stays entirely inside the box.
To see another step in the process of further reducing the vertical extent,
width, of the box click on the button below.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
9/22
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Two Sided Nature of Limits
We must be able to make f (x ) arbitrarily close to L by making x sufficiently
close to a, on either side of a.
In some cases we get a different value if we make x close to a on different
sides of a. In this case the limit does not exist.
This leads to the idea of one-sided limits. We will introduce a function in
which one-sided limits play a role in Example 3 and then give the definition
of one-sided limits.
There are other cases in which a limit does not exist. We will investigate
some of them later.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
10/22
Example 3 – A Step Function
Consider the function
g (x ) =
y
|x |
x
1
x
−1
Clint Lee
Math 112 Lecture 1: Introduction to Limits
11/22
The Absolute Value Function (A digression)
The absolute value function is
another piecewise function. Its
formula is
|x | =

 −x

if
0 if
x if
y
x <0
x =0
x >0
x
Its graph looks like this:
Clint Lee
Math 112 Lecture 1: Introduction to Limits
12/22
Continuing Example 3
For the function g defined at the beginning of Example 3, what can we say
about the limit
lim g (x )?
x →0
Clint Lee
Math 112 Lecture 1: Introduction to Limits
13/22
Definition of One-Sided Limits
As seen in Example 3 a function can get close to different values as the
limit point a is approached from the two different sides. This leads to:
Definition (Limits from the Right and Left)
We say that
if f (x ) can be made arbitrarily close to R by making x sufficiently close to a
with
, that is for
. This is the limit of f at a from the
right, or from above, and
if f (x ) can be made arbitrarily close to L by making x sufficiently close to a
with
, that is for
. This is the limit of f at a from the left,
or from below.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
14/22
Checking for the Existence of a Limit
We can check to see if a limit exists by checking the one-sided limits.
Existence of a Limit
The limit
lim f (x )
x →a
exists and is equal to L if and only if both
lim f (x )
x →a−
and
lim f (x )
x →a+
, and
lim f (x ) =
x →a−
and
lim f (x ) =
x →a+
Otherwise, the regular, two-sided, limit does not exist.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
15/22
Example 4 – Calculating Limits
y
Consider the function
6


2x + 8 if
−2x + 4 if
h (x ) =
 2
x − 2x + 2 if
x < −1
−1 ≤ x < 1
x ≥1
4
2
Graph each branch separately:
−2
Clint Lee
Math 112 Lecture 1: Introduction to Limits
2
x
16/22
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explain why.
(a)
(c)
(e)
lim
x →−1−
h (x )
(b)
lim h (x )
(d)
lim h (x )
(f)
x →−1
x → 1+
Clint Lee
lim
x →−1+
h (x )
lim h (x )
x → 1−
lim h (x )
x →1
Math 112 Lecture 1: Introduction to Limits
17/22
Infinite Limits
Consider the function
f (x ) =
x +1
x2 − 1
from Example 1. This function takes arbitrarily large, positive and negative,
values near x = 1. So the function is undefined there. We can use limits to
say this more precisely.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
18/22
Definition of Infinite Limits
Definition (Infinite Limits)
We say that
lim f (x ) = ∞
x →a
if f (x ) can be made arbitrarily
close to a, on either side of a,
by making x sufficiently
Similar definitions apply for one-sided infinite limits.
Clint Lee
Math 112 Lecture 1: Introduction to Limits
19/22
Example 5 – Evaluating Infinite Limits
For the function
f (x ) =
x +1
x2 − 1
from Example 1 give the value as +∞ or −∞ for each limit, or explain why
the infinite limit does not exist.
(a)
(b)
(c)
lim f (x )
x → 1−
lim f (x )
x → 1+
lim f (x )
x →1
Clint Lee
Math 112 Lecture 1: Introduction to Limits
20/22