微积分第二章课件

lim f (x) L
xa
if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x greater than a.
7
Similarly, we get definition of the right-hand limit of f(x) as x approaches a.
1
We choose x 1, then
mPQ
x2 1 x 1
. For instance, for
the point Q(1.5, 2.25) we have
mPQ
2.25 1 1.5 1
2.5
The closer Q is to P, the closer x is to 1 and the closer
lim g(x) 3, lim g(x) 1
x2
x2
lim g(x) 2, lim g(x) 2
x5
x5
We have lim g(x) 2 g(5). x5
Figure 5
9
x
x
For instance, since lim 1 lim 1, therefore
x
x x0
x x0
3
In order to understand the precise meaning of a function
in Definition , let us begin to consider the behavior of a
function
0.5x 1, x 1
f (x) as x approaches 1.
0,
x 1
From the graph of f shown in Figure 2, we can intuitively
see that as x gets closer to 1 from both sides but x≠1, f(x) gets closer to 3/2. In this case, we use the notation
xa
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.
mPQ is to 2. This suggests that
Q
the slope of the tangent
line t should be m =2. P
Figure 1
2
2.2 Limits of Functions 2.2.1 Limit of a Function f(x) as x Approaches a Definition 1 Let f be a function defined on some open
lim does not exist.
x0 x
Example 5 Suppose that
f
(x)
1 2
x2,
x 0,
x 1, x 0.
(1) Find lim f (x) and lim f (x).
x0
(2) Discuss
lim
f (x).
x0
x0
Solution (1) lim f (x) 1, lim f (x) 0 .
x0
x0
(2) Because lim f (x) 1≠ lim f (x) 0, so lim f (x)
x0
x0
x0
does not exist.
10
Example 6 Show that lim x 0. x0
Solution Recall that
We have
x if x 0 x x if x 0
Figure 3
5
Example 3 The Heaviside function H is defined by
H
(t)
0 1
if t 0 if t 0
As t approaches 0 from the left, H(t) approaches 0. As t
approaches 0 from the right, H(t) approaches 1. There is
xa
xa
xa
14
Examples of these four cases are given in Figure 7.
2x
2x
lim lim
x3 x 3
x3 x 3
lim tan x
x( / 2)
Figure 7
lim ln x
x0
15
Definition 5 The line x=a is called a vertical asymptote of
to a.
As an example we have
lim
x0源自文库
2 x2
.
Similar definitions can be given for the one-sided
infinite limits
lim f (x) lim f (x) lim f (x) lim f (x)
xa
x x0
x x0
lim f (x) L if and only if lim f (x) lim f (x) L.
xa
xa
xa
Example 4 Use the graph of y=g(x) to find the following limits, if they exist.
Solution This graph shows that
-2 -1
2。
1
.1
2
-1 Figure 2
-2
lim f (x) 3 2,
x1
and say that the limit of f(x), as x approaches 1, is 3/2, or that f(x)
approaches 3/2 as x approaches 1.
4
Example 2 Guess the value of lim sin x .
x0 x
SOLUTION The function f(x)=sinx/x is not defined at x=0. From the table and the graph in Figure 3 we guess that
lim sin x 1 x0 x
This guess is in fact correct, as will be proved in Chapter 3.
no single number that H(t) approaches as t approaches 0.
Figure 4
6
2.2.2 One-Sided Limits Definition 2 Let f be a function defined on an open interval of the form (a, c) for some real number c, and let L be a real number. We say that the right-hand limit of f(x) as x approaches a from the right is L, and write
interval containing a except possibly at a itself, and let L be a real number. We say that the limit of f(x) as x approaches a is L, and write
lim f (x) L
a, except possibly at a itself. Then lim f (x) xa
means that the value of f(x) can be made arbitrarily large
negative by taking x sufficiently close to a, but not equal
SOLUTION As x becomes close to 0, x2 also becomes
close to 0, and 1/x2 becomes very large. (See the table on
the next page.)
12
It appears from the graph of the function f(x) shown in Figure that the value of the f(x) can be made arbitrarily
xa
means that the value of f(x) can be made arbitrarily large by
taking x sufficiently close to a, but not equal to a.
Example 6
Find
1
lim
x0
x2
if it exists.
Chapter 2 Limits and Derivatives
2.1The tangent and velocity problems 2.1.1 The tangent problem Example 1 Find an equation of the tangent line to the parabola y x2 at the point P(1,1). SOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty is that we know only one point , P, on t, whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby Q(x,x2) on the parabola and computing the slope mPQ of the secant line PQ.
lim x lim(x) 0 and lim x lim x 0
x0
x0
x0
x0
Therefore,
lim x 0.
x0
11
2.2.3 Infinite Limits
Definition 3 Let f be a function defined on both sides of a, except possibly at a itself. Then lim f (x)
the curve y=f(x) if at least one of the following statements
x
1 0.5 0.2 0.1 0.01 0.001
1 x2
1 4 25 100 10, 000 1, 000, 000
large by taking x close enough to 0. Thus
lim
x0
1 x2
Figure 6
13
Definition 4 Let f be a function defined on both sides of
xa
if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a.
8
x
x
For example, lim 1, lim 1.
Let f be a function defined on an open interval of the form (c, a) for some real number c, and let L be a real number. We say that the left-hand limit of f(x) as x approaches a from the left is L, and write lim f (x) L