第5.1节 无限极限

5.1 INFINITE LIMITS

Up to this point we have studied three types of limits:

The limit notation ____________________ means that whenever H is positive infinite, f(H) ≈ L (Figure 5.1.1(a)).

__________________ means that whenever x ≈ c and x ≠ c, f(x) is negative infinite (Figure 5.1.1(b)). The various other combinations have the meanings which one would expect.

EXAMPLE 1 ________________.

EXAMPLE 2 ____________________.

EXAMPLE 3 Find ___________________

Let H be positive infinite. Then

and therefore

Thus the limit exists and is _____.

Figure 5.1.1

EXAMPLE 4 Find ___________________

We have x³ + 200 x² = x² (x+200). When H is negative infinite, H ² is positive infinite

and (H+200) is negative infinite, so their product is negative infinite. Thus

______________________.

When ___________________

The limit does not exist, because f(x) has no standard part. The infinity symbol is only used to indicate the behavior of f(x) and is not to be construed as a number.

EXAMPLE 5

A student can get a score of 100t / (t+1) on his math exam if he studies t hours for it (Figure 5.1.2). If the studies infinitely long for the exam, his score will be infinitely close to 100, because if H is positive infinite,

In the notation of limits,

__________________________.

Figure 5.1.2

EXAMPLE 6 Given any polynomial

______________________

Of degree n > 0, the limits as t approaches -∞ or +∞ are as follows.

Suppose an > 0. When n is even, __________ f(t) = ∞ , _______________

When n is odd, _____________ ________________.

The signs are all reversed when an< 0.

All these limits can be computed from

EXAMPLE 7 In the special theory of relativity, a body which is moving at constant velocity

v, -c< v < c, will have mass

and its length in the direction of motion will be

_____________________________

Here m0, l0, and c are positive constants denoting the mass at rest (that is, the mass when v= 0), the length at rest, and the speed of light.

Suppose the velocity v is infinitely close to the speed of light c, that is,

v= c- ε, ε > 0 infinitesimal.

Then

which is the square root of a positive infinitesimal. Thus _________________ is a positive infinitesimal. Therefore for v infinitely close to c, m is positive infinite and l is positive infinitesimal. That is, a body moving at velocity infinitely close to (but less than) the speed of light has infinite mass and infinitesimal length in the direction of motion. In the notation of limits this means that

Caution: This example must be understood in the light of our policy of speaking as if a line in physical space really is like the hyperreal line. Actually, there is no evidence one way or the other on whether a line in space is like the hyperreal line, but the hyperreal line is a useful model for the purpose of applications.

EXAMPLE 8 Evaluate ________________

When H is positive infinite, sin H is between - 1 and 1 and thus finite, so (sin H) / H is

infinitesimal. The limit is therefore zero:

_____________________.

EXAMPLE 9 Find ____________

If H is any integer or hyperinteger, then

cos (2πH) = 1, cos ((2πH + π ) = -1.

In fact, cos x will keep oscillating between 1 and -1 even for infinite x. Therefore the limit does not exist.

Limits involving ex and 1n x will be studied in Chapter 8.

PROBLEMS FOR SECTION 5.1

Find the following limits. Your answer should be a real number, ∞, -∞, or “ does not exist.”

With a calculator, compute some value as x approaches its limit, and see what happens.

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69 Prove that if ___________________ then ____________ .

71 Prove that if __________ and f(x) >0 for all x, then __________ .

72 Prove that if __________ exists or is infinite, then

_______________________

73 Prove that if _____________exists or is infinite then

______________________________