Download Calculus I: Section 1.3 Intuitive Limits

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Calculus I: Section 1.3
Intuitive Limits
Intuitive Definition. We write
lim f (x) = L
x→a
and say, “The limit of f (x) as x approaches a equals L,” if we can make the values of f (x) arbitrarily close
to L by taking x to be sufficiently close to a on either side of a but not equal to a.
This says that the values of f (x) tend to come closer to L as x comes closer and closer to a from either
side of a, but x 6= a. It is very important to remember that when working with limits, we never consider
x = a. Here is alternative notation which we may see:
f (x) → L
Example 1.
as
x → a.
Guess the value of the limit:
x−1
.
x→1 x2 − 1
lim
It will be useful to have the following tables of values, which you can get on your calculators or computers
while doing HW:
x<1
0.5
0.9
0.99
0.999
0.9999
f (x)
0.666667
0.526316
0.502513
0.500250
0.500025
x>1
1.5
1.1
1.01
1.001
1.0001
f (x)
0.400000
0.476190
0.497512
0.499750
0.4999975
By examining these tables, we can see that as x comes closer and closer to 1 from above and from below,
the value of the function approaches to
. Therefore, we conclude that
lim
x→1
x−1
=
x2 − 1
.
What is f (1)?
Notice that lim f (x) 6= f (1) in this example. That is not always the case, but it is noteworthy.
x→1
1
Important!
In order for a limit to exist, f (x) must approach the
same value as x approaches a from both sides.
Example 2.
The Heaviside function is defined by
(
0
H(t) =
1
if t < 0
if t ≥ 0
)
.
Find limt→0 H(t), if it exists.
This function is easy to graph. When we know the graph of a function, we can use it to help us find
the limit of a function as it approaches a particular x-value.
This example brings us to the concept of one-sided limits. As t approaches 0 from the right, H(t)
approaches 1. As t approaches 0 from the left, H(t) approaches 0. The limit itself does not exist because
these values are not the same. However, they are still meaningful.
We write
lim− f (x) = L
x→a
and say that the left-hand limit of f (x) as x approaches a is equal to L if we can make the values of f (x)
arbitrarily close to L by taking x to be sufficiently close to a while x < a.
Similarly, we write
lim+ f (x) = L
x→a
to represent the right-hand limit. The definition is the same with the exception that x > a.
Therefore we have lim H(t) = 0 and lim H(t) = 1.
t→0−
t→0+
2
Important!
In order for a limit to exist, f (x) must approach the
same value as x approaches a from both sides.
Example 2.
The Heaviside function is defined by
(
0
H(t) =
1
if t < 0
if t ≥ 0
)
.
Find limt→0 H(t), if it exists.
This function is easy to graph. When we know the graph of a function, we can use it to help us find
the limit of a function as it approaches a particular x-value.
This example brings us to the concept of one-sided limits. As t approaches 0 from the right, H(t)
approaches 1. As t approaches 0 from the left, H(t) approaches 0. The limit itself does not exist because
these values are not the same. However, they are still meaningful.
We write
lim− f (x) = L
x→a
and say that the left-hand limit of f (x) as x approaches a is equal to L if we can make the values of f (x)
arbitrarily close to L by taking x to be sufficiently close to a while x < a.
Similarly, we write
lim+ f (x) = L
x→a
to represent the right-hand limit. The definition is the same with the exception that x > a.
Therefore we have lim H(t) = 0 and lim H(t) = 1.
t→0−
t→0+
2
Important!
In order for a limit to exist, f (x) must approach the
same value as x approaches a from both sides.
Example 2.
The Heaviside function is defined by
(
0
H(t) =
1
if t < 0
if t ≥ 0
)
.
Find limt→0 H(t), if it exists.
This function is easy to graph. When we know the graph of a function, we can use it to help us find
the limit of a function as it approaches a particular x-value.
This example brings us to the concept of one-sided limits. As t approaches 0 from the right, H(t)
approaches 1. As t approaches 0 from the left, H(t) approaches 0. The limit itself does not exist because
these values are not the same. However, they are still meaningful.
We write
lim− f (x) = L
x→a
and say that the left-hand limit of f (x) as x approaches a is equal to L if we can make the values of f (x)
arbitrarily close to L by taking x to be sufficiently close to a while x < a.
Similarly, we write
lim+ f (x) = L
x→a
to represent the right-hand limit. The definition is the same with the exception that x > a.
Therefore we have lim H(t) = 0 and lim H(t) = 1.
t→0−
t→0+
2