Section 1.5
Introduction to Limits
In the last section, we predicted the value for the slope of a tangent line by investigating slopes
of secant lines. This idea leads directly to the idea of a limit. Informally, we will think of a limit as
a prediction of a function’s value at a point x = a based on how the function behaves near x = a.
For example, think about the two pictures below.
Notice that other than the obvious color difference, the two graphs look exactly alike. In each
graph, we know everything about how the functions behave, except for how they behave at x = 3.
The black line through the graphs is acting like a blindfold, so we have no idea what f (3) or g(3) is.
But even though we don’t know the values for f (3) and g(3), we can use the information we have
about the curves to make a guess as to the values. Inspecting the red curve f (x), I would hazard
a guess that f (3) should be 8. Similarly, it appears that g(3) should be 8 as well. We have just
(informally) evaluated our first limits. We inspected the behavior of f (x) near the input x = 3,
and predicted that the function’s value at this input would be 8. We formalize this by writing
lim f (x) = 8.
x→3
We drew a similar conclusion about g(x), and so we can also write
lim g(x) = 8.
x→3
Let’s record a formal definition for a limit:
Definition 1. Suppose that f (x) is defined when x is near the number a. We write
lim f (x) = L
x→a
to mean that we can make the values for f (x) as close as we like to L by choosing inputs x sufficiently
close to a.
If no such number L exists, we say that the limit of f (x) as x → a does not exist (DNE).
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Section 1.5
There is something important to notice here. As a limit at x = a is simply a prediction of a
function’s value at x = a, its actual value at x = a (i.e. f (x)) does not affect the limit. In other
words,
lim f (x) may not be the same as f (a).
x→a
Let’s go back to our previous example to try to understand this. We decided that
lim f (x) = 8 and lim g(x) = 8,
x→3
x→3
but I was sneaky and didn’t let you see the whole graph of either function. The actual graphs are
below:
Notice that the red curve f (x) behaved exactly the way that we predicted–in other words,
lim f (x) = 8 = f (3).
x→3
On the other hand, the blue curve g(x) did not behave as we expected it to. In this case,
lim g(x) = 8 but g(3) = 3.
x→3
Example. Use the graph below to find values for the following:
f (−2)
lim f (x)
x→−2
f (−1)
lim f (x)
x→−1
f (0)
lim f (x)
x→0
f (2)
lim f (x)
x→2
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Section 1.5
The dot at the point (−2, 0) indicates that f (−2) = 0. However, at x values near x = −2, the
function is very close to −9; so
lim f (x) = −9.
x→−2
From the graph, it is clear that f (−1) = −5. To determine the limit of f (x) as x → −1, we
inspect the way the graph behaves at inputs near x = −1; in this case, the function approaches
−5, and so the limit is −5:
lim f (x) = −5.
x→−1
Notice that there is an open circle at x = 0, which means that f (0) does not exist. However,
we can still try to evaluate limit of f (x) as x → 0: at x values close to 0, f (x) gets close to 1, so
lim f (x) = 1.
x→0
Finally, let’s think about the input x = 2: the dot indicates that f (2) = 6, but near x = 2, the
function takes on values close to −5, so
lim f (x) = −5.
x→2
The data is recorded below:
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Section 1.5
lim f (x) = −9
f (−2) = 0
x→−2
f (−1) = −5
lim f (x) = −5
x→−1
f (0) DNE
lim f (x) = 1
x→0
lim f (x) = −5
f (2) = 6
x→2
Example. Let
f (x) =
x2 − 3x + 2
.
x−1
Use numerical methods to find
lim f (x).
x→1
Keep in mind that f (1) does not have to be the same as limx→1 f (x). In fact, we know that
this particular function is not even defined at x = 1: when we attempt to evaluate f (1), we get a
fraction whose denominator is 0, i.e. f (1) does not exist.
However, we can still attempt to evaluate the limit of f (x) as x approaches 1. In other words,
we will predict how the function ought to behave at x = 1 by investigating its behavior near x = 1.
To do this, let’s fill out the chart below:
x f (x)
0
.5
.9
1.1
1.5
2
All we need to do to fill in the chart is to evaluate f (x) at these points. We’ll use a calculator
to fill in the chart:
x f (x)
0
-2
.5
-1.5
.9
-1.1
1.1
-.9
1.5
-.5
0
2
The points are plotted below:
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Section 1.5
Notice that, as the x values for the points approach 1, the y values for the points appear
approach −1. So we say that
x2 − 3x + 2
lim
= −1.
x→1
x−1
The graph of f (x) is plotted below, and confirms our calculations:
One-sided Limits
Many interesting situations can occur with function limits. For example, let’s think about the
limit of f (x) as x → 1 in the graph below:
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Section 1.5
If we were only to consider the left-hand side of the graph, where x < 1, we would definitely say
that the limit of f (x) as x → 1 is 1. However, if we only look at the right-hand side, where x > 1,
we would say that the limit of f (x) as x → 1 is 3.
Because of this inconsistency, the limit of f (x) as x → 1 does not exist. However, this answer is
a bit unsatisfying; it doesn’t capture the behavior that we noticed above. For this reason, we will
define left and right-hand limits below:
Definition 2. We write
lim f (x) = L
x→a−
and say that the limit of f (x) as x → a from the left is L, if we can make the function values f (x)
as close as we like to L by choosing inputs x sufficiently close to a and less than a.
Similarly, we write
lim f (x) = M
x→b+
and say that the limit of f (x) as x → b from the right is M , if we can make the function values
f (x) as close as we like to M by choosing inputs x sufficiently close to b and greater than b.
Example. In light of these new definitions, let’s return to the graph we considered a moment ago:
Again, we know that
lim f (x) DNE .
x→1
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Section 1.5
However, we have already shown that both the left-hand and right-hand limits exist: we write
lim f (x) = 1 and lim f (x) = 3.
x→1−
x→1+
These ideas lead us to a quick observation–right and left hand limits must exist and be equal in
order for the general limit to exist. In other words,
lim f (x) = L if and only if lim f (x) = L and lim f (x) = L.
x→a−
x→a
x→a+
If the right-hand and left-hand limits do not match up, then we may immediately conclude that
the general limit
lim f (x)
x→a
does not exist.
Infinite Limits The graph below displays another interesting phenomenon that we may observe
when attempting to evaluate limits. Consider the limit of f (x) = x12 as x → 0:
It appears that, the closer x gets to 0, the larger the outputs of f (x) become. This is another
case in which the limit does not exist, as the outputs f (x) are not approaching any specific value.
Again, we write
lim f (x) DNE.
x→0
However, as with the last example, this answer is not particularly satisfying. We can say more
about the limit of f (x) as x → 0–in particular, we can specify the way in which it fails to exist.
Thus we make the following definition:
Definition 4. Let f (x) be a function that is defined on both sides of x = a, except perhaps at a
itself. Then we write
lim f (x) = ∞
x→a
to indicate that the limit of f (x) as x → a does not exist because the values for f (x) can be made
arbitrarily large (as large as we like) by choosing x sufficiently close to (but not equal to) a.
Similarly,
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Section 1.5
Definition 5. Let f (x) be a function that is defined on both sides of x = a, except perhaps at a
itself. Then we write
lim f (x) = −∞
x→a
to indicate that the limit of f (x) as x → a does not exist because the values for f (x) can be made
to be arbitrarily large negative numbers by choosing x sufficiently close to (but not equal to) a.
An important point to make here is that when we write
lim f (x) = ∞,
x→a
we are saying that the limit of f (x) as x → a does not exist–we are simply specifying why the limit
does not exist.
Example. Using the graph below, evaluate the limit of f (x) = x12 as x → 0.
We have already concluded that this limit does not exist; now we can specify why it doesn’t exist.
Since the function’s outputs f (x) can be made as large as we like by choosing x to be sufficiently
close to 0, we write
lim f (x) = ∞.
x→0
One quick note here–notice that the vertical line x = 0 is a vertical asymptote of f (x) = x12 ,
since f approaches the line but never crosses it.
Example. Using the graph below, evaluate the limit of g(x) = x1 as x → 0.
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Section 1.5
In this example, it is again clear that the indicated limit does not exist. However, we can’t treat
the limit the same way that we did in the example above: on the left-hand side of 0, the function
values are approaching −∞, while on the right-hand side, the function values are approaching ∞.
So for the general limit, we must content ourselves with saying that
lim g(x) DNE.
x→0
However, we can specify left-hand and right-hand limits: the discussion above tells us that
lim g(x) = −∞ and lim g(x) = ∞.
x→0−
x→0+
Notice that, as with the last example, the vertical line x = 0 is a vertical asymptote of f (x) = x1 .
You probably recognize that asymptotes are very closely tied to infinite limits. Indeed, we can
actually use limits to define the vertical asymptotes of a function:
Definition 6. The line x = a is a vertical asymptote of the function f (x) if any of the following
statements is true:
lim f (x) = ∞
x→a
lim f (x) = −∞
x→a
lim f (x) = ∞
x→a−
lim f (x) = −∞
x→a−
lim f (x) = ∞
x→a−
lim f (x) = −∞
x→a−
Below, we summarize different ways in which a limit might fail to exist:
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Section 1.5
1. The right-hand and left-hand limits do not match up, as in the graph below, where the limit
of f (x) as x → 1 DNE:
In a case such as this, we should try to compute the left-hand and right-hand limits.
2. The function values approach infinity (or negative infinity) as in the graph below, where the
limit of f (x) as x → 1 DNE:
In this case, we note write that the limit of f (x) is ∞.
3. The function values ”bounce around” and never approach a specific number, as in the graph
below, where we can only say that the limit of f (x) as x → 0 DNE:
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Section 1.5
This particular function, sin(1/x), is known as the Topologist’s Sine Curve.
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