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onesidedlimits - One-Sided Limits In the final two examples...

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One-Sided Limits In the final two examples in the previous section we saw two limits that did not exist. However, the reason for each of the limits not existing was different for each of the examples. We saw that did not exist because the function did not settle down to a single value as t approached . The closer to we moved the more wildly the function oscillated and in order for a limit to exist the function must settle down to a single value. However we saw that did not exist not because the function didn’t settle down to a single number as we moved in towards , but instead because it settled into two different numbers depending on which side of we were on. In this case the function was a very well behaved function, unlike the first function. The only problem was that, as we approached , the function was moving in towards different numbers on each side. We would like a way to differentiate between these two examples. We do this with one-sided limits . As the name implies, with one-sided limits we will only be looking at one side of the point in question. Here are the definitions for the two one sided limits. Right-handed limit
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We say provided we can make f(x) as close to L as we want for all x sufficiently close to a and x>a without actually letting x be a . Left-handed limit We say provided we can make f(x) as close to L as we want for all x sufficiently close to a and x<a without actually letting x be a . Note that the change in notation is very minor and in fact might be missed if you aren’t paying attention. The only difference is the bit that is under the “lim” part of the limit. For the right-handed limit we now have (note the “+”) which means that we know will only look at x>a . Likewise for the left-handed limit we have (note the “-”) which means that we will only be looking at x<a .
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