Limits At Infinity

Limits At Infinity - Limits At Infinity, Part I In the...

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Limits At Infinity, Part I In the previous section we saw limits that were infinity and it’s now time to take a look at limits at infinity. By limits at infinity we mean one of the following two limits. In other words, we are going to be looking at what happens to a function if we let x get very large in either the positive or negative sense. Also, as we’ll soon see, these limits may also have infinity as a value. First, let’s note that the set of Facts from Infinite Limit section also hold if the replace the with or . The proof of this is nearly identical to the proof of the original set of facts with only minor modifications to handle the change in the limit and so is left to the reader. We won’t need these facts much over the next couple of sections but they will be required on occasion. In fact, many of the limits that we’re going to be looking at we will need the following two facts. Fact 1 1. If r is a positive rational number and c is any real number then, 2. If r is a positive rational number, c is any real number and x r is defined for then, The first part of this fact should make sense if you think about it. Because we are requiring we know that x r will stay in the denominator. Next as we increase x then x r will also increase. So, we have a constant divided by an increasingly large number and so the result will be increasingly small. Or, in the limit we will get zero.
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The second part is nearly identical except we need to worry about x r being defined for negative x . This condition is here to avoid cases such as . If this r were allowed then we’d be taking the square root of negative numbers which would be complex and we want to avoid that at this level. Note as well that the sign of c will not affect the answer. Regardless of the sign of c we’ll still have a constant divided by a very large number which will result in a very small number and the larger x get the smaller the fraction gets. The sign of c will affect which direction the fraction approaches zero ( i.e. from the positive or negative side) but it still approaches zero. If you think about it this is really a special case of the last Fact from the Facts in the previous section. However, to see a direct proof of this fact see the Proof of Various Limit Properties section in the Extras chapter. Let’s start the off the examples with one that will lead us to a nice idea that we’ll use on a regular basis about limits at infinity for polynomials. Example 1 Evaluate each of the following limits. (a) [ Solution ] (b) [ Solution ] Solution (a) Our first thought here is probably to just “plug” infinity into the polynomial and “evaluate” each term to determine the value of the limit. It is pretty simple to see what each term will do in the limit and so this seems like an obvious step, especially since we’ve been doing that for other limits in previous sections. So, let’s see what we get if we do that. As
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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Limits At Infinity - Limits At Infinity, Part I In the...

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