Transcript x - mor media international

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Introduction to Limits
Limits—Notation and Definition
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Definition of the limit of a function f(x) at a number c:
Let f(x) be a function such that as values of x are getting closer to a constant c from both left
and right, but remains unequal to c, the corresponding function values of f(x) (i.e., the yvalues) are getting closer to a real number N, we say the limit of f(x) as x is approaching c
exists and this limit is N. In notation, we write limx→c f(x) = N or lim f(x) = N.
x→c
If no such N exist, we say the limit does not exist, or simply DNE.
As we can see, the limit of a function is not about the entire function, but rather, it is just about
the function at a particular number: When x is approaching this number, what number does the
function value f(x) (i.e., y) approach? Let’s look at the following examples below:
1.
2.
3
3.
3
2
3
1
2
lim f(x) =
2
lim f(x) =
lim f(x) =
x→2
x→2
x→2
4.
5.
lim f(x) =
x→2
2
lim f(x) =
x→2
2
One-Sided Limits
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As you recall, when we talk about the limit of a function f(x) as x approaches c, we have to
look at the values of x approaching c from both left and right. However, we can look at the
values of x approaching c from one side only—this is called one-sided limit.
If we only look at the values of x approaching c from the left side only, it’s called the leftsided limit, and similarly, if we only look at the values of x approaching c from the right side
only, it’s called the right-sided limit.
1.
lim f(x) =
3
x→2–
Left-sided limit: lim– f(x) = L
x→c
lim f(x) =
2
x→2+
Definition: As values of x are getting closer
lim f(x) =
and closer to c from the left, but remains less
x→2
than c, the corresponding function values of
f(x) (i.e., the y-values) are getting closer and
lim f(x) =
2.
3
x→2–
closer to a real number L.
1
lim f(x) =
Right-sided limit: lim+ f(x) = R
x→c
2
x→2+
lim f(x) =
x→2
Definition: As values of x are getting closer
and closer to c from the right, but remains Properties:
greater than c, the corresponding function 1. If limxc– f(x) = limxc+ f(x), then limxc f(x)
exists and __________________________.
values of f(x) (i.e., the y-values) are getting
closer and closer to a real number R.
2. If limxc– f(x)  limxc+ f(x), then limxc f(x)
__________________.
Continuity—At a Number
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As you might recall, we say a function is continuous everywhere if there are no ____ or
_____. However, this is only an informal definition. Let’s present you here with the formal
definition of continuity, however, not for the entire function, but rather at one number.
Definition of f(x) is continuous at a number c:
We say f(x) is continuous at x = c, if all the following properties are satisfied:
1. f(c) is defined, i.e., f(c) must be some real number.
2. limxc– f(x) = f(c), i.e., the left-sided limit must be same as f(c).
3. limxc+ f(x) = f(c), i.e., the right-sided limit must be same as f(c).
In other words, a function f(x) is continuous at c if
limxc f(x) = f(c),
i.e., the limit of the function at c must be same as the function value at c.
If one (or more) of the three properties fails, we say f(x) is discontinuous at c. In other words,
if limxc f(x)  f(c), then f(x) is discontinuous at c.
Obviously, f(x) is discontinuous at x = _____________
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x The Three Properties
2
–4: (1)
(2)
(3)
1
–1: (1)
(2)
(3)
1: (1)
(2)
(3)
–6 –5 –4 –3 –2 –1 0
–1
3: (1)
(2)
(3)
5: (1)
(2)
(3)
–2
8: (1)
(2)
(3)
–3
1
2
3
4
5
6
7
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Continuity—On All Real Numbers
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Definition of f(x) is continuous everywhere:
We say a function f(x) is continuous on all real numbers (or everywhere) if there are no
points of discontinuity.
Some functions are continuous everywhere while others are not.
Examples of functions that are continuous everywhere:
1.
2.
3.
4.
Examples of functions that are not continuous everywhere:
1.
2.
3.
4.
Continuity—On an Interval
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We see that not all functions are continuous on all real numbers (i.e., everywhere). However,
we can still talk about the continuity of function even if it is not continuous everywhere.
Definition of f(x) is continuous on an open interval and closed interval
We say a function f(x) is continuous on an open interval of (a, b) if
there are no points of discontinuity on the interval (a, b). That is, f(x)
is continuous at any x where a < x < b.
We say a function f(x) is continuous on a closed interval of [a, b] if
i. f(x) is continuous on the open interval (a, b),
ii. f(a) and f(b) are defined, and
ii. limxa+ f(x) = f(a) and limxb– f(x) = f(b).
3
2
1
–6 –5 –4 –3 –2 –1 0
–1
1
2
3
4
5
6
7
8
9
10
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–2
–3
Since f(x) is discontinuous at x = _____________,
so f(x) is continuous on the intervals: ________________________
a
b
a
b