Infinite Limits and Asymptotes
Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.
Limits As X Becomes Arbitrarily Large
The same type of questions we considered about a function as approached a finite number can also asked about as "becomes arbitrarily large," "increases without bound," and is eventually larger than any fixed number.
Example 1: What happens to the values of (Fig. 1) and as becomes arbitrarily large, as increases without bound?
Solution: One approach is numerical: evaluate and for some "large" values of and see if there is a pattern to the values of and . Fig. 1 shows the values of and for several large values of . When is very large, it appears that the values of are close to and the values of are close to . In fact, we can guarantee that the values of are as close to as someone wants by taking to be "big enough".
Fig. 1
The values of may or may not ever equal (they never do), but if is "large," then is "close to" . Similarly, we can guarantee that the values of are as close to as someone wants by taking to be "big enough". The graphs of and are shown in Fig. 2 for "large" values of .
Fig. 2
Practice 1: What happens to the values of and as becomes arbitrarily large?
The answers for Example 1 can be written as limit statements:
"As becomes arbitrarily large, the values of approach can be written "" and "the values of approach ". can be written " ".
The symbol " " is read "the limit as approaches infinity" and means "the limit as becomes, arbitrarily large" or as increases without bound. (During this discussion and throughout this book, we do not treat "infinity" or "", as a number, but only as a useful notation. "Infinity" is not part of the real number system, and we use the common notation " " and the phrase " approaches infinity" only to mean that " becomes arbitrarily large". The notation " , " read as " approaches negative infinity," means that the values of become arbitrarily large.)
Practice 2: Write your answers to Practice 1 using the limit notation.
The asks about the behavior of as the values of get larger and larger without any bound, and one way to determine this behavior is to look at the values of at some values of which are "large". If the values of the function get arbitrarily close to a single number as gets larger and larger, then we will say that number is the limit of the function as approaches infinity. A definition of the limit as " " is given at the end of this section.
Practice 3: Fill in the table in Fig. 3 for and , and then use those values to estimate and .
Fig. 3
Example 2: How large does need to be to guarantee that (assume )?
Solution: If , then (Fig.4). If , then .
In general, if is any positive number, then we can guarantee that by picking only values of : if , then .
From this we can conclude that .
Fig. 4
Practice 4: How large does need to be to guarantee that (assume )? Evaluate
The Main Limit Theorem (Section 1.2) about limits of combinations of functions is true if the limits as " " are replaced with limits as " ", but we will not prove those results.
Polynomials occur commonly, and we often need the limit, as , of ratios of polynomials or functions containing powers of . In those situations the following technique is often helpful:
(i) factor the highest power of in the denominator from both the numerator and the denominator, and
(ii) cancel the common factor from the numerator and denominator.
The limit of the new denominator is a constant, so the limit of the resulting ratio is easier to determine.
Solutions:
Similarly,
If we have a difficult limit, as , it is often useful to algebraically manipulate the function into the form of a ratio and then use the previous technique.
If the values of the function oscillate and do not approach a single number as becomes arbitrarily large, then the function does not have a limit as approaches infinity: the limit Does Not Exist.
Solution: and do not have limits as . As grows without bound, the values of oscillate between and (Fig. 5), and these values of do not approach a single number. Similarly, continues to take on values between and , and these values are not approaching a single number.
Fig. 5
Using Calculators To Help Find Limits as " " or " "
Calculators only store a limited number of digits of a number, and this is a severe limitation when we are dealing with extremely large numbers.
Example: The value of is clearly equal to for all values of , and your calculator will give the right answer if you use it to evaluate or . Now use it to evaluate for a big value of , say . , but most calculators do not store digits of a number, and they will respond that which is wrong. In this example the calculator's error is obvious, but the same type of errors can occur in less obvious ways when very large numbers are used on calculators.
You need to be careful with and somewhat suspicious of the answers your calculator gives you.
Calculators can still be helpful for examining some limits as " " and " " as long as we do not place too much faith in their responses.
Even if you have forgotten some of the properties of natural logarithm function and the cube root function , a little experimentation on your calculator can help you determine that .