2. Limits at Infinity

Transcription

1 2 Limits at Infinity To understand sequences and series fully, we will need to have a better understanding of its at infinity We begin with a few examples to motivate our discussion EXAMPLE 1 Find SOLUTION Both the numerator and denominator of the fraction are approaching infinity Unfortunately, this doesn't tell us anything about the it it depends on how the two infinities compare In the language of calculus, is an indeterminate form Let's try plugging in some values for For large values of approaches,,,,,, the numerator is almost exactly twice the denominator, so the ratio What's happening is that the constant terms ( in the numerator and in the denominator) are becoming less and less important as grows larger As a result, these constants play no role in the outcome: As a general rule, finding a it as often has to do with figuring out which parts of a formula you can ignore The trick is to concentrate on the largest term of any sum, ignoring the smaller terms For a polynomial, this the highest power of dominant term is whichever term has the highest degree, ie

2 THEOREM (LIMITS OF RATIONAL FUNCTIONS) Let and be nonzero polynomials, and let and be their respective dominant terms Then EXAMPLE 2 Evaluate the following its (a) (b) (c) SOLUTION We use the above theorem in each case (a) The dominant term in the numerator is the the Thus, and the dominant term in the denominator is (c) The dominant term in the numerator is the the Thus, and the dominant term in the denominator is (c) The dominant term in the numerator is the dominant term in the denominator is the, and thus (including the negative sign) The Our goal is to expand upon this method by evaluating the sizes of different quantities as The ideas discussed here are the beginning of a field of math known as asymptotic analysis, and are used throughout mathematics and science (especially physics and computer science) to evaluate the rate at which quantities grow or shrink as

3 The Asymptotic Hierarchy We begin by defining precisely what it means for one quantity to be much smaller than another DEFINITION OF Let and be sequences We say that is much smaller than as, and write as if The symbol, which means is much smaller than, consists of two less than symbols ( ) in a row We will also use the symbol for is much larger than, where means the same thing as EXAMPLE 3 For a basic example, observe that is much smaller than as, written as This is because Intuitively, the statement as expresses the idea that infinity squared is much larger than infinity More generally, we have the following rule THEOREM (COMPARING POWERS) If, then as This rule even works in the case where and are non-integers For example, since as

4 EXAMPLE 4 or is larger as? How do polynomials compare with exponentials? For example, do you think that The following table compares these sequences for different values of,,,,,,,,,,,,,,,,,,,,,,,,,,,, As you can see, grows much more quickly than, so as In the last example, it may surprise you that is so much larger than The idea is that multiplication is very powerful: is larger because it is the product of different things, while is the product of only five things This reasoning applies whenever you are comparing an exponential with a polynomial THEOREM (EXPONENTIALS VS POWERS) For any value of and any, we have as For example, it turns out that is much smaller than as Though gets off to a slow start, it overtakes starting at,, and grows much more quickly than from that point on Incidentally, the rule for comparing exponentials is very simple THEOREM (COMPARING EXPONENTIALS) If, then as

5 EXAMPLE 5 How do logarithms compare with polynomials and exponentials? If you think about it, log actually grows very slowly as For example, here is a table comparing and log for different vales of log,,,,,, As you can see, log as It turns out that log grows more slowly than any power of For example, log is much smaller than (the hundredth root of ) for large values of THEOREM (LOGARITHMS VS POWERS) For any, we have log as Note that logarithms with different bases are simply multiples of one another For example, ln log log log Thus, the base of a logarithm doesn't affect its size very much EXAMPLE 6 So far, the largest quantities we have found are the exponentials: Is anything larger than all of these? Yes Recall that the factorial of (written ) is the product of all numbers from to : It is not hard to see that must be larger than For example, here is a comparison of and factorial:

6 Both are the product of factors, but in the case of most of the factors are much larger than For this same reason, will be larger than for any base Much larger than even is the quantity For example, is clearly much larger than For comparison, is a 31-digit number, is a 158-digit number, and is a 201-digit number We now have a fairly clear picture of the hierarchy of functions THE ASYMPTOTIC HIERARCHY ln powers of exponentials Knowing the positions of functions in this hierarchy can make certain its very easy to evaluate THEOREM (LIMITS OF RATIOS) Let and be sequences (a) If as, then (b) If and are positive and as, then EXAMPLE 7 Evaluate the following its ln (a) (b) (c) SOLUTION Limit (a) is since ln as Limit (b) is also since as Finally, it (c) is since and are both positive and as The following theorem lists a few more important properties of

7 THEOREM (PROPERTIES OF ) Let,, and be sequences, and let and be nonzero constants 1 If and as, then as 2 If as, then as Part (2) of this theorem essentially says that you can ignore coefficients when comparing the sizes of expressions For example, since log log as We are now ready to define a more general notion of dominant terms DEFINITION (DOMINANT TERMS) A term of a sum is dominant if it is much larger than every other term EXAMPLE 8 Find the dominant term in each of the following sums (a) (b) (c) log SOLUTION The dominant term in (a) is the The dominant term in (b) is the The dominant term in (c) is the (Note that, since ) Asymptotic Equivalence In many cases, it is only necessary to consider the dominant term in a sum For example, in a quotient of two polynomials, we can replace each polynomial by its dominant term when computing the it: The idea is that the original numerator is roughly the same as just, since the doesn't contribute very much Similarly, the original denominator is roughly the same as just the other two terms are much smaller, since The following definition makes precise the idea that two quantities are roughly the same

8 DEFINITION (ASYMPTOTIC EQUIVALENCE) Two sequences and are asymptotically equivalent, written as if For example, is asymptotically equivalent to, since More generally: Any sum is asymptotically equivalent to its dominant term For example, is asymptotically equivalent to, since The following theorem lets us use asymptotic equivalence to compute its THEOREM (EVALUATING LIMITS) 1 Asymptotically equivalent sequences have the same it That is, if as, then 2 If and as, then EXAMPLE 9 To compute the it, ln observe that and ln Using part (2) of the previous theorem,

9 we conclude that ln and therefore, using part (1) of the previous theorem,, ln Of course, you don't need to give this much detail when you're doing calculations Just keep in mind that replacing the numerator or denominator of a fraction with its dominant term is safe ln EXAMPLE 10 Find SOLUTION The is the dominant term in the numerator, and the is the dominant term in the denominator Therefore, ln Further Techniques There are a few more cases where asymptotic techniques may be used to evaluate its THEOREM (ANALYZING POWERS) Let and be sequences If as, then for any constant power In particular, as as

10 EXAMPLE 11 Evaluate SOLUTION Since, it follows that Then EXAMPLE 12 Evaluate SOLUTION Since, it follows that Then THEOREM (ANALYZING SIZES) Let,, and be sequences, where and as If then as well as as This theorem says that we can use asymptotic equivalence when analyzing sizes of terms For example, we can use this theorem to conclude that ln The reason is that the left side is asymptotically equivalent to, the right side is asymptotically equivalent to, and

11 EXAMPLE 13 Find SOLUTION Observe that, which means that is the dominant term in the numerator Then THEOREM (ANALYZING PRODUCTS) Let,, and be sequences, where and as Then as EXAMPLE 14 Find SOLUTION Observe that and It follows that so, THEOREM (ANALYZING LOGARITHMS) Let and be sequences, where as, and suppose that Then ln ln as EXAMPLE 15 Find ln SOLUTION Since, it follows that ln ln ln

12 Then ln ln ln Pitfalls There are a few important pitfalls to avoid when reasoning asymptotically EXAMPLE 16 Consider the it It might seem that the under the first square root isn't contributing much, so that? But this is not correct: the actual value of the it is, as illustrated in the following table,,,, The problem here is that you can't ignore smaller terms in a sum if the big terms cancel In this case, what's happening is that is only approximately, which means that is only approximately Indeed, some smaller stuff When we subtract, what's left over isn't zero it's the smaller stuff! In this case, the smaller stuff is getting closer and closer to By the way, there is a nice way to evaluate this it algebraically The strategy is to multiply the numerator and denominator by the conjugate of the radical expression:

13 Since, the denominator is asymptotically equivalent to, and therefore the it is EXAMPLE 17 Consider the it Clearly the denominator is asymptotically equivalent to What about the numerator? You might be tempted to change the to an but this is not correct! The reason is that the exponent in an exponential matters a lot Small changes in an exponent can affect the size of an expression a lot In general, if and is a constant, it does not follow that The key to evaluating this it is to leave the as it is: By the way, the same warning holds for factorials: just because does not mean that As with exponents, the inside of a factorial expression matters a lot, and it is not safe to ignore small terms Hardy vs Landau The and notation we have been using is known as Hardy notation, named after British mathematician G H Hardy It is popular is some branches of mathematics as well as physics As it happens, there is a competing notation for the concepts known as Landau notation, named for German mathematician Edmund Landau The Landau notation is popular in certain other branches of mathematics as well as computer science The following table compares the two notations

14 Hardy Notation Landau Notation Here would be pronounced little oh of sub, and would be pronounced Theta of sub We will not be using the Landau notation, but you ought to be aware that it exists

15 EXERCISES 1 8 Evaluate the it Evaluate the it 1 2 ln ln ln ln ln 9 14 Evaluate the it 19 ln ln 9 20 ln ln ln 12 ln Evaluate the it