Date: 11/24/97 at 07:52:20 From: Susan Batchilder Subject: Calculus: limits In my textbook, the limit of (1/x) as x approaches zero is defined as infinity. But the limit as you approach zero from the negative side is negative infinity and the limit as you approach zero from the positive side is positive infinity. Therefore since the limit from the right does not equal the limit from the left, then by definition limit should not exist. So, why does my textbook have the limit defined as infinity? I look forward to having the mystery solved. Thank you, Susan
Date: 11/24/97 at 12:32:16 From: Doctor Rob Subject: Re: Calculus: limits Dear Susan, Your reasoning showing that the limit does not exist is correct. Because the limit does not exist, saying that it "equals infinity" is, on the face of it, nonsensical. This function behaves in a special way that makes the limit not exist. The way it behaves is that as x gets small in absolute value, the function gets large in absolute value. Stated more technically, given any number N > 0, we can find a d > 0 such that whenever |x| < d, then |1/x| > N. The value of d happens to be 1/N for this particular function, but would be different for other functions. This means that as x gets small in absolute value, the function grows without bound in absolute value. The function is said to have a vertical asymptote at x = 0, and the common phrase that the limit "equals infinity" is actually a shorthand for this situation. So this is why your textbook might have said that the limit was infinity. But why is this phrase chosen? It has to do with convenience. Mathematicians like the fact that limits, when they exist, commute with all the operations of arithmetic. We want this property to hold true for all limits, and when we try to extend it to include limits of functions like 1/x as x -> 0, we end up with exactly the same structure as when we try to append "infinity" to the usual real numbers, and the same difficulties. Examples of these difficulties are infinity minus infinity not being definable, infinity over infinity not being definable, and infinity times zero not being definable. Even though working with infinity as a number gives rise to all of the difficulties above, it is still easier than working with something that's "undefined." There are other ways that limits might not exist, but do not "equal infinity," or have their absolute values get arbitrarily large. One example is the function f(x) defined by: f(x) = -1 if x < 0 f(x) = 0 if x = 0 f(x) = +1 if x > 0. f(x) is a function whose limit as x approaches zero does not exist, because there is a jump at 0. A more difficult example is the function: sin(1/x) as x -> 0 for which the limit also doesn't exist, due to ever-increasingly rapid oscillation. In neither of these situations do we use the phrase that the limit "equals infinity." I hope this makes sense. If not, write again. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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