Finding the Limit
Date: 10/24/95 at 21:30:14 From: Anonymous Subject: Definition of limit For a limit to exist, the right side and left side have to equal each other. Say you have a limit such as "lim as x->4 for function (3x-4). To see if this function exists, you have to check it on both sides. This is where my question is. Would you plug in 3 for the left and 5 for the right? If I do that I would not get the same answer. So what am I doing wrong? I have asked my Calculus teacher and she said that was not it but she did not explain how. I have looked in several calculus books to try to teach myself, but the examples are not that good. Please help.
Date: 11/5/95 at 16:1:2 From: Doctor Jeremy Subject: Re: Definition of limit The thing that you're confused about is what "left" and "right" mean there. They are a special kind of limit. The "left" limit is the limit as you get closer to 4 from the left (below). The "right" limit is the limit as you get closer to 4 from the right (above). You want those limits to be the same or the limit doesn't really exist. In this case, the limit on the left is 8: if x=3 then (3x-4)=5 if x=3.9 then (3x-4)=7.7 if x=3.99 then (3x-4)=7.97 The limit on the right is also 8: if x=5 then (3x-4)=11 if x=4.1 then (3x-4)=8.3 if x=4.01 then (3x-4)=8.03 So it gets closer to 8 as x gets closer to 4 on _either side_, meaning the limit exists and is 8. Sometimes you can get right and left limits, but they don't match up so you don't have a limit. Here's an example: Let f(x) be 1 if x is greater than or equal to zero, and f(x) be 0 if x is less than zero. Then the limit of f(x) as x->0 on the left is 0, since whenever x is negative, no matter how close to zero, f(x) is 0. But the limit on the right is 1, since whenever x is positive, no matter how small, f(x) is 1. So lim as x->0 f(x) doesn't exist, since the left and right limits don't match up. But there are no particular numbers you can pick (such as you tried with 3 and 5) to get the "left" and "right". -Doctor Jeremy, The Math Forum
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