computinglimits

# computinglimits - Computing Limits In the previous section...

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Computing Limits In the previous section we saw that there is a large class of function that allows us to use to compute limits. However, there are also many limits for which this won’t work easily. The purpose of this section is to develop techniques for dealing with some of these limits that will not allow us to just use this fact. Let’s first got back and take a look at one of the first limits that we looked at and compute its exact value and verify our guess for the limit. Example 1 Evaluate the following limit. Solution First let’s notice that if we try to plug in we get, So, we can’t just plug in to evaluate the limit. So, we’re going to have to do something else. The first thing that we should always do when evaluating limits is to simplify the function as much as possible. In this case that means factoring both the numerator and denominator. Doing this gives,

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So, upon factoring we saw that we could cancel an from both the numerator and the denominator. Upon doing this we now have a new rational expression that we can plug into because we lost the division by zero problem. Therefore, the limit is, Note that this is in fact what we guessed the limit to be. On a side note, the 0/0 we initially got in the previous example is called an indeterminate form . This means that we don’t really know what it will be until we do some more work. Typically zero in the denominator means it’s undefined. However that will only be true if the numerator isn’t also zero. Also, zero in the numerator usually means that the fraction is zero, unless the denominator is also zero. Likewise anything divided by itself is 1, unless we’re talking about zero. So, there are really three competing “rules” here and it’s not clear which one will win out. It’s also possible that none of them will win out and we will get something totally different from undefined, zero, or one. We might, for instance, get a value of 4 out of this, to pick a number completely at random. There are many more kinds of indeterminate forms and we will be discussing indeterminate forms at length in the next chapter. Let’s take a look at a couple of more examples. Example 2 Evaluate the following limit. Solution
In this case we also get 0/0 and factoring is not really an option. However, there is still some simplification that we can do. So, upon multiplying out the first term we get a little cancellation and now notice that we can

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computinglimits - Computing Limits In the previous section...

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