The Limit of a Function

33  Download (0)

Full text

(1)

The Limit of a Function

To find the tangent to a curve or the velocity of an object, we now turn our attention to limits in general and numerical and graphical methods for computing them.

Let’s investigate the behavior of the function f defined by f(x) = x2 – x + 2 for values of x near 2.

(2)

The Limit of a Function

The following table gives values of f(x) for values of x close to 2 but not equal to 2.

(3)

From the table and the graph of f (a parabola) shown in Figure 1 we see that the closer x is to 2 (on either side of 2), the closer f(x) is to 4.

The Limit of a Function

(4)

The Limit of a Function

In fact, it appears that we can make the values of f(x) as close as we like to 4 by taking x sufficiently close to 2.

We express this by saying “the limit of the function f(x) = x2 – x + 2 as x approaches 2 is equal to 4.”

The notation for this is

(5)

The Limit of a Function

In general, we use the following notation.

This says that the values of f(x) approach L as x

approaches a. In other words, the values of f(x) tend to get closer and closer to the number L as x gets closer and

(6)

The Limit of a Function

An alternative notation for

is f(x) → L as x → a

which is usually read “f(x) approaches L as x approaches a.”

Notice the phrase “but x ≠ a” in the definition of limit. This means that in finding the limit of f(x) as x approaches a, we never consider x = a. In fact, f(x) need not even be defined when x = a. The only thing that matters is how f is defined near a.

(7)

The Limit of a Function

Figure 2 shows the graphs of three functions. Note that in part (c), f(a) is not defined and in part (b), f(a) ≠ L.

But in each case, regardless of what happens at a, it is true that limx→a f(x) = L.

(8)

Example 1

Guess the value of Solution:

Notice that the function f(x) = (x – 1)/(x2 – 1) is not defined when x = 1, but that doesn’t matter because the definition of limx→a f(x) says that we consider values of x that are close to a but not equal to a.

(9)

Example 1 – Solution

The tables below give values of f(x) (correct to six decimal places) for values of x that approach 1 (but are not equal to 1).

On the basis of the values in the tables, we make the guess that

cont’d

(10)

The Limit of a Function

Example 1 is illustrated by the graph of f in Figure 3.

Now let’s change f slightly by giving it the value 2 when x = 1 and calling the resulting function g:

Figure 3

(11)

The Limit of a Function

This new function g still has the same limit as x approaches 1. (See Figure 4.)

(12)

One-Sided Limits

(13)

Example 6

The Heaviside function H is defined by

[This function is named after the electrical engineer Oliver Heaviside (1850–1925) and can be used to describe an electric current that is switched on at time t = 0.] Its graph is shown in Figure 8.

(14)

Example 6

As t approaches 0 from the left, H(t) approaches 0. As t

approaches 0 from the right, H(t) approaches 1. There is no single number that H(t) approaches as t approaches 0.

Therefore does not exist.

cont’d

(15)

One-Sided Limits

We noticed in Example 6 that H(t) approaches 0 as t approaches 0 from the left and H(t) approaches 1 as t approaches 0 from the right.

We indicate this situation symbolically by writing and

(16)

One-Sided Limits

The notation t → 0 indicates that we consider only values of t that are less than 0.

Likewise, t → 0+ indicates that we consider only values of t that are greater than 0.

(17)

One-Sided Limits

Notice that Definition 2 differs from Definition 1 only in that we require x to be less than a.

(18)

One-Sided Limits

Similarly, if we require that x be greater than a, we get “the right-hand limit of f (x) as x approaches a is equal to L”

and we write

Thus the notation x → a+ means that we consider only x greater than a. These definitions are illustrated in Figure 9.

Figure 9

(19)

One-Sided Limits

By comparing Definition 1 with the definitions of one-sided limits, we see that the following is true.

(20)

Example 7

The graph of a function g is shown in Figure 10. Use it to state the values (if they exist) of the following:

Figure 10

(21)

Example 7 – Solution

From the graph we see that the values of g(x) approach 3 as x approaches 2 from the left, but they approach 1 as x approaches 2 from the right.

Therefore

and

(c) Since the left and right limits are different, we conclude from (3) that limx→2 g(x) does not exist.

(22)

Example 7 – Solution

The graph also shows that and

(f) This time the left and right limits are the same and so, by (3), we have

Despite this fact, notice that g(5) ≠ 2.

cont’d

(23)

Infinite Limits

(24)

Infinite Limits

Another notation for limx→a f(x) = ∞ is f(x) → ∞ as x → a

(25)

Infinite Limits

Again, the symbol ∞ is not a number, but the expression limx→a f(x) = ∞ is often read as

“the limit of f(x), as x approaches a, is infinity”

or “f(x) becomes infinite as x approaches a”

or “f(x) increases without bound as x approaches a”

(26)

Infinite Limits

This definition is illustrated graphically in Figure 12.

Figure 12

(27)

Infinite Limits

A similar sort of limit, for functions that become large

negative as x gets close to a, is defined in Definition 5 and is illustrated in Figure 13.

(28)

Infinite Limits

The symbol limx→a f(x) = –∞ can be read as “the limit of f(x), as x approaches a, is negative infinity” or “f(x)

decreases without bound as x approaches a.” As an example we have

(29)

Infinite Limits

Similar definitions can be given for the one-sided infinite limits

remembering that x → a means that we consider only

values of x that are less than a, and similarly x → a+ means that we consider only x > a.

(30)

Infinite Limits

Illustrations of these four cases are given in Figure 14.

(31)

Infinite Limits

(32)

Example 10

Find the vertical asymptotes of f(x) = tan x.

Solution:

Because

there are potential vertical asymptotes where cos x = 0.

In fact, since cos x → 0+ as x → (π/2) and cos x → 0 as x → (π/2)+, whereas sin x is positive (near 1) when x is near π/2, we have

(33)

Example 10 – Solution

This shows that the line x = π/2 is a vertical asymptote.

Similar reasoning shows that the lines x = π/2 + nπ, where n is an integer, are all vertical asymptotes of f(x) = tan x.

The graph in Figure 16 confirms this.

cont’d

Figure

Updating...

References

Related subjects :