CHAPTER 2. LIMITS
2.2
25
Limits
Recall one of the main conclusions of the previous section: that sometimes we do a
calculation, have one of the inputs of the calculation get closer and closer to a single
number, and see that outputs are getting closer and closer to a single number. If we
take this idea, and make a formal definition out of it, we get the technical concept
of limit.
Definition. We write


the y-values of f (x) become infinitely close to L
as the x-values become infinitely close, (but not


equal) to the number a.
“ lim f (x) = L” to mean that
x→a
Essentially the idea of limit is this: keep track of what happens to the outputs
as the inputs are getting close to some specific value. The hard part is: (1) we are
keeping track of both inputs and outputs at the same time, and (2) that we’re not
just plugging in one input, but keeping track of all the inputs that are “close” to
some specific value.
Note two things about this definition. (1) The phrase “infinitely close” is vague.
But it can be made precise using a somewhat technical mathematical description
involving finite, but arbitrarily small, distances (δ’s and ε’s). (2) When we have x
infinitely close to a, we do not let x equal a.
Example 1. Example 1 from section 2.1 involved a limit. Can you figure out what
we took the limit of and what it equalled?
Solution. Although we started with a function p(t), the height of a ball, this wasn’t
what we took the limit of. If you don’t see the limit right away, think about this:
what sort of t-values were we plugging in? Were they getting closer and closer to
something? Hopefully you see that we took lim of something. What were we
t→2.3
plugging these t-values into? Not just p(t), although that was part of it. We were
p(t) − p(2.3)
plugging them into
. Putting all this together, we have the following:
t − 2.3
p(t) − p(2.3)
Example 1 in section 2.1 showed that lim
≈ −19.04.
t→2.3
t − 2.3
Example 2. Example 2 from Section 2.1 involved a limit. Can you figure out what
we took the limit of and what it equalled?
Solution. Although we started with a function sin(x), this wasn’t what we took
the limit of. If you don’t see the limit right away, think about this: what sort of
x-values were we plugging in? Were they getting closer and closer to something?
Hopefully you see that we took lim of something. What were we plugging these
x→0.8
x-values into? Not just sin(x), although that was part of it. We were plugging them
sin(x) − sin(0.8)
into
. Putting all this together, we have the following
x − 0.8
sin(x) − sin(0.8)
Example 2 in Section 2.1 showed that lim
≈ 0.7.
x→0.8
x − 0.8
sin(x − 3)
.
x→3
x−3
Example 3. Find lim
CHAPTER 2. LIMITS
26
Solution. At this point, the only way we can find this limit, is to use our calculators.
We will use a table of numbers.
x
2.5
2.9
2.99
3
3.5
3.1
3.01
sin(x − 3)
x−3
0.95885
0.99833
0.99998
DNE
0.95885
0.99833
0.99998
To read the limit from this table you have to look at two things at the same time: if
the y-values are near something fixed number when the x-values are near 3. If you
do this, I hope you see that the y-values are getting close to y = 1 as the x-values
get close to x = 3. Thus, the limit is 1
lim
x→3
sin(x − 3)
=1
x−3
sin(x − 3)
. What
x−3
you should look at is what the y-values are doing on the graph for those x-values
near 3.
You can confirm that this looks right by looking at a graph of
I hope that you see the same thing as in the table: it appears that the y-values are
getting infinitely close to 1.
Example 4. Find the following limit, if it exists.
lim sin
x→1
�
1
x−1
�
CHAPTER 2. LIMITS
27
Solution. Again, we make a table of numbers
�
�
1
x sin
x−1
0.5 −0.90930
0.9 0.54402
0.99 0.50637
1.5 0.90930
1.1 −0.54402
1.01 −0.50637
Maybe you’d guess that the real answer is half-way between 0.506 and −0.506. So
that’d be zero, but it’s not very convincing. Maybe we should look at more numbers:
0.999
0.99999
1.001
1.0001
−0.82688
−0.03575
0.82688
−0.30561
Well, it’s still not very clear what’s happening, after all, −0.035 and −0.305 aren’t
that close, relatively speaking.
In this case, a graph is much more useful than a table of numbers. The reason
is twofold: (1) a graph shows us a lot more numbers at once than the handful in
our tables, and (2) our brains see patterns better in graphs; we see curves, and
movement and geometry in the shapes of the graph. These patterns may reveal
what is going on.
Here’s the graph, although it is perhaps confusing:
What’s happening is that the graph oscillates up and down between 1 and −1 faster
and faster as x approaches 1. Therefore, there is not a single number that all the
y-values get closer and closer to, as x gets closer to 1. Therefore, this limit does not
exist
�
�
1
= DNE
lim sin
x→1
x−1
This is where we ended on Tuesday, September 10
CHAPTER 2. LIMITS
28
Definition. We write
“ lim f (x) = L” to mean that
x→a+
We write
“ lim f (x) = L” to mean that
x→a−
We write
“ lim f (x) = ∞” to mean that
x→a


the y-values of f (x) become infinitely close to L as
the x-values become infinitely close to the number


a, but with x > a.


the y-values of f (x) become infinitely close to L as
the x-values become infinitely close to the number


a, but with x < a.
�
�
the y-values of f (x) become infinitely large as the
x-values become infinitely close (but not equal) to
the number a.
We can also combine these concepts to have
lim f (x) = ∞, lim f (x) = ∞, lim f (x) = −∞, lim f (x) = −∞, lim f (x) = −∞.
x→a+
x→a
x→a−
x→a+
x→a−
Example 5. Find the following limits, if they exist:
lim
x→3+
x|6 − 2x|
,
3−x
lim
x→3−
x|6 − 2x|
,
3−x
x|6 − 2x|
.
x→3 3 − x
lim
Solution. We do this problem graphically.
From looking at the graph we conclude that the limits
lim = −6,
x→3+
Example 6. Find lim
x→2
lim = 6,
x→3−
lim = DNE.
x→3
1
.
(x − 2)2
Solution. We graph the function, with two different windows, in case there is a
peak that we can’t see in the first window.
CHAPTER 2. LIMITS
29
From these graphs we see that the function has a vertical asymptote at x = 2. The
best way to describe this is to write
1
=∞
x→2 (x − 2)2
lim
Note: people also sometimes say that this limit does not exist. It is correct to say
this, because the limit is not a real number, however it is better to write ∞ as we
have, because it provides more information. If you like, you can write both DNE
and ∞.
Example 7. Let f (x) be the function defined by the graph below. Find the following, if they exist.
(a)
(b)
lim f (x)
(e) lim f (x)
lim f (x)
(f) f (1)
(g) lim f (x)
x→−2−
x→−2+
(c) f (−2)
(d) lim f (x)
x→1−
x→1+
x→3−
(h) lim f (x)
x→3+
(i) f (3)
CHAPTER 2. LIMITS
30
Solution. Here’s the short version, followed by explanations.
(a)
(b)
(e) lim f (x) = ∞
lim f (x) = 3
x→−2−
x→1+
(f) f (1) DNE
(g) lim f (x) = 2
lim f (x) = 3
x→−2+
(c) f (−2) DNE
(d) lim f (x) = ∞
x→3−
(h) lim f (x) = 4
x→3+
x→1−
(i) f (3) = 5
(a) Look at the graph, near x = −2, but only to the left of −2. What happens
to the y-values as x gets closer to −2 (but only to the left of −2). The y-values get
closer to 3. In other words,
lim f (x) = 3.
x→2−
(b) Look at the graph, near x = −2, but only to the right of −2. What happens
to the y-values as x gets closer to −2 (but only to the right of −2). The y-values
get closer to 3. In other words,
lim f (x) = 3.
x→2+
(c) f (−2) is not defined: this is what the open circle at x = −2 means.
(d) Look at the graph, near x = 1, but only to the left of 1. What happens to
the y-values as x gets closer to 1 (but only to the left of 1). The y-values get larger
and larger. They get infinitely large. In other words,
lim f (x) = ∞.
x→1−
(e) Look at the graph, near x = 1, but only to the right of 1. What happens
to the y-values as x gets closer to 1 (but only to the right of 1). The y-values get
larger and larger. They get infinitely large. In other words,
lim f (x) = ∞.
x→1+
(f) f (1) is not defined. There is a vertical asymptote at x = 1, that’s what the
dashed line means.
(g) Look at the graph, near x = 3, but only to the left of 3. What happens to
the y-values as x gets closer to 3 (but only to the left of 3). The y-values get closer
to 2. In other words,
lim f (x) = 2.
x→3−
(h) Look at the graph, near x = 3, but only to the right of 3. What happens
to the y-values as x gets closer to 3 (but only to the right of 3). The y-values get
closer to 4. In other words,
lim f (x) = 4.
x→3+
(i) f (3) = 5. This is what the black circle at the point (3, 5) means.
This is where we ended on Wednesday, September 11
Example 8. Make up a graph of a function f (x) that has the following properties:
CHAPTER 2. LIMITS
•
•
lim f (x) = −3
x→−3−
lim f (x) = −3
x→−3+
• f (−3) = 3
31
• lim f (x) = −3
x→5−
• lim f (x) = 1
x→5+
• f (5) = 3
Solution. You can’t make up a single, basic formula that has all these properties.
But the definition of “limit” means that we only need a graph that has these properties very close to certain points, in this case near x = −3 and x = 5. The trick is
that you can make up any graph you want away from these points. In the graphs
below, basically the only part that matters is right around the three open circles,
and the two black circles. Here are two solutions, one minimalistic and one overly
complicated.