第5.3节 极限与曲线绘制

5.3 LIMITS AND CURVE SKETCHING

By definition, _______________ means that for every hyperreal number x which is infinitely close but not equal to c, f(x) is infinitely close to L. What does ____________ tell us about f(x) for real numbers x? It turns out that if _________, then for every real number x which is close to but not equal to c, f(x) is close to L.

In the next section we shall justify the above intuitive statement by a mathematical theorem. The main difficulty is to make the word “close” precise. For the time being we shall simply illustrate the idea with some examples.

EXAMPLE 1 Consider the limit_______________

This limit is evaluated by letting x ≠ 0 be infinitesimal:

_________________

_____________________

Let us see what happens if instead of taking x to be infinitely small we take x to be a “small” real number. We shall make a table of values of

_________________________

for various small x.

We see that as x gets closer and closer to zero, f(x) gets closer and closer to 2.

With a calculator, the student should try this for some of the limits on pages 124 and 241.

The table helps us to draw the graph of the curve y=f(x). Although the point (0,2) is not on the graph, we know that when x is close to 0. f(x) is close to 2, and draw the graph accordingly. The graph is drawn in Figure 5.3.1.

Other types of limits also give information which is useful in drawing graphs. For instance, if _____________ , then for every number x which is close to but not equal to c, the value of f(x) is large. And if _________________ , then for every large real number x, f(x) is close to L.

In both the above statements, if we replace “close” by “infinitely close” and “large” by “infinitely large” we get our official definition of a limit. We give two more examples.

Figure 5.3.1

EXAMPLE 2 Consider the limit ________________________

For x infinitely close but not equal to 2, 1/ (x-2)² is positive infinite. Let us make a table of values when x is a real number close to but not equal to 2.

As x gets closer and closer to 2, f(x) gets larger and larger.

EXAMPLE 3 _____________________

For infinitely large x, 1 + 1/ (x-2)² is infinitely close to 1. Here is a table of values

of 1+1/ (x-2)² for large real x.

As x gets large, 1+1/ (x-2)² gets close to 1. Also notice that

_______________________

and for large negative x, 1+1/(x-2)² is close to 1.

In Chapter 3 we showed how to use the first and second derivatives to sketch the graph of a function which is continuous on a closed interval. In the next example we shall sketch the graph of the function f(x)=1+1/ (x-2)². But this time the function is discontinuous at x = 2, and the domain is the whole real line except for the point x=2. Our method uses not only the values but also the limits of the function and its first derivative.

EXAMPLE 4 Sketch the curve _________________________

The first two derivatives are

f ′(x) = -2 (x-2) -3 f ′′(x) = 6(x-2) -4.

The first and second derivatives are never zero. f(x) is undefined at x=2. In our table we shall show the values of f(x) and its first two derivatives at a point on each side of x=2. We shall also show the limit of f(x) and its first derivative as x→ -∞, x→ 2 -, x→ 2+, and x→ ∞. (we will not need the limits of f ′′(x).)

The first line of the table, ________________ shows that for large negative x the curve is close to 1 and its slope is nearly horizontal. The second line, x=1, shows that the curve is increasing and concave upward in the interval( -∞, 2), and passes through the point (1, 2) with a slope of 2. The third line, _____________, shows that just before x=2 the curve is far above the x-axis its slope is nearly vertical. Going through the table in this way, we are able to sketch the curves as in Figure 5.3.2.

The curve approaches the dotted horizontal line y = 1 and the dotted vertical line x = 2. These lines are called asymptotes of the curve.

Figure 5.3.2

Suppose the function f and its derivative f ′ exist and are continuous at all but a finite number of points of an interval 1. The following procedure can be used in sketching the curve y=f(x).

Step 1 First carry out the procedure outlined in Section 3.9 concerning the first and second

derivative.

Step 2 Compute ___________ and ____________.

(They may either be real numbers, + ∞, -∞, or may not exist.)

Step 3 At each point c of I where f is discontinuous, compute f(c), ________and __________.

( Some or all of these quantities may be undefined.)

Step 4 Compute ___________________ and __________________.

Step 5 At each point where f ′ is discontinuous. compute f(c), ____________ and ________.

We shall now work several more examples; the steps in computing the limits are left to the student.

EXAMPLE 5 f(x) = x 3/5.

Then ________________, ___________________.

At the point x=0, f(x)=0 and f ′(x) does not exist. We first plot a few points, compute the

necessary limits, and make a table.

Figure 5.3.3 is a sketch of the curve.

Figure 5.3.3

The behavior as x approaches -∞, ∞, and zero are described by the limits we have computed. As x approaches either -∞ or ∞, f(x) gets large but the slope becomes more nearly horizontal. As x approaches zero the curve becomes nearly vertical, increasing from left to right, so we have a vertical tangent line at x = 0.

EXAMPLE 6 f(x) = x 4/5

Then ________________, ___________________.

f ′(x) is undefined at x = 0. We make the table:

With this information we can sketch the curve in Figure 5.3.4.

Figure 5.3.4

This time the limits of the derivative as x approaches zero show that there is a cusp at x = 0, with the curve decreasing when x< 0 and increasing when x >0.

EXAMPLE 7 Sketch the curve __________ for 0< x < 2π.

f(x) and f ′(x) are undefined at x = π because the denominator sin π is zero. The first two derivatives are

_______________, _____________________.

Thus f ′ (x) is always negative, and f ′′(x) =0 when x= π/2, 3π/2. Here is the table:

Notice that the table from π to 2π is just a repeat of the table from 0 to π.

This is because

_____________________

The curve is sketched in Figure 5.3.5.

Figure 5.3.5

PROBLEMS FOR SECTION 5.3

1 This figure is a sketch of a curve y=f(x). At which points x=c do the following happen?

(a) f is discontinuous at c

(b) _______ does not exist

(c) ________ does not exist

(d) f is not differentiable at c

(e) ______ does not exist

(f) _______does not exist.

In Problems 2-42, sketch the graph of f(x). Use a table of values of f(x), f ′(x), f ′′(x), and limits of f(x) and f ′(x). Then check your answer by using a graphics calculator to draw the graph.

2 _____________ 3 f(x) = x² - 2x

4 f(x)= x³-x 5 ___________

6 ____________ 7 ___________

8 9

10 11

12 13

14 15

16 17

18 f(x) = 2 - (x-1) 1/3 19 ____________________

20 __________________ 21 __________________

22 __________________ 23

24 25

26 27

28 29

30 31

32 33

34 35

36 37

38 39

40 41

42

In Problems 43-55, graph the given function.

43 f(x) =|x| -1 44 f(x) = 1-|2x|

45 f(x) =|2x -1| 46 _____________

47 f(x) =2x+| x-2| 48 f(x) =x² +| x |

49 f(x) =x² +| x + 1 | 50 f(x) =|x² - 1 |

51 _____________ 52 f(x) =x/ |x|

53 _____________ 54 ____________

□55 ________________