Division by Zero in Limits

# Division by Zero in Limits - Division by 0 in Limits: What...

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Division by 0 in Limits: What to do with 0 + (an addendum to Chapter 2.2 of Stewart's Calculus ) A Typical Example of an Unfortunate Division by 0: Let's start examining the role of 0 limits in division by examining the limit: Here, the 3 + means that we're looking at x values as they approach “from the right” or “from above”. That is, values of x greater than 3. As always in trying to compute limits on rational functions, we'll attempt to plug in the value x is approaching in place of the variable: I've stepped outside the usual notation by putting the sign is in quotes, since clearly our result is nonsense. We hoped we would get a value, which would be limit, and equality would hold. But since it has failed, our method did not work, so it cannot possibly be equal. You might think this is the end of the calculation, that we cannot compute a result, but actually, because of how the values are approaching x = 3, there is a little bit more information which we can use to work out an answer. The limit will NOT exist, in the strictest sense, but since this is a limit, we aren't really dividing by zero, only approaching it, we CAN say something about that approach. Since x 3 + , we know that x > 3. So we're only considering values ever so slightly more positive than 3. In the past, we might have considered plugging in a value such as 2.9999, but this is always a bit risky: Merely plugging in a value we might still pick a number too far away from 3. Our function might twist and curve so rapidly that this approximate value might give false results. Not to mention that if we don't have a calculator, we might be too hard to calculate anything with a number 2.9999 anyway! Instead we'll just plug in “ 3 +” for x to act as a placeholder for a number “slightly” more positive than our value, 3. Page #1 of 7 3 2 lim 3 x x x  3 23 2 1 lim " " ?!? 33 3 0 x x x  

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But what does this mean? Well 1 + is just very near 1, but ever so slightly more positive. Nothing special about that. And as we approach our limit, this will just become closer and closer until we actually get a 1 value. But just being near 0 in the denominator, specifically being just slightly positive, makes a very big difference to our outcome.
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## This note was uploaded on 12/14/2010 for the course MATH 1za3 taught by Professor Ben during the Winter '10 term at Macalester.

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Division by Zero in Limits - Division by 0 in Limits: What...

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