Figuring Out Limits

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rayroshi

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I have a rather general questions about limits. Often, it seems that a limit problem appears to be just a no-brainer, and could be solved by simple inspection, then substitution ("plug and chug" method); however, just as often, that turns out to not be the case. For example, after I look at the problem, seeing that, as X in the denominator approaches infinity, the answer should be zero, so I blithely move on to the next problem, only to find out, later, that I was dead wrong. So it appears that one can't simply take a cursory glance at a problem and simply assume that the answer can be found by inspection. Hence, my question: Is there a general rule of thumb/technique/method that one can use to tell if a limit problem can be solved by simple inspection followed by the 'plug and chug' or not? (Note: I have purposely avoided giving an actual example, here, because I wanted to avoid having someone simply solving that particular problem; i.e., I am more interested in learning a general approach to solving limits than learning the answer to any particular problem, as there must be something about limits that I'm not understanding.

Any help would be greatly appreciated.
 
The general rule is, don't just glance at a problem and decide to skip it because it looks easy; actually carry out the substitution and see what happens!

You have learned already that cursory glances are not enough, because tricky cases can hide well. Apply that lesson!

Maybe you'll want to show us an example of one that you thought didn't require any thought, and tell us why. The general rule is simple; actually applying it is not ...
 
Thanks for your reply.

However, I think you didn't read what I wrote carefully enough. I didn't say that I skipped anything, but rather, after looking at the some of those problems, (for example) it appeared as though substituting infinity would simply result in an answer of zero...which turns out to be wrong. So I would then not skip the problem, but rather submit zero as an answer, since it seemed to be reasonable and logical--obvious, even--only to find out that what seemed obvious was not right (so much for 'obvious'). As you so correctly stated, it seems that applying the general rule is not easy. What I am seeking is just what those general rules are, so I can see why what I am doing is wrong.

Additionally, as I said, I didn't want to include any specific example, hoping, instead, to learn some general rules, rather than just seeing someone work out the answer to a problem. There are tons of problems worked out on the internet, of course, but that doesn't help me see the real reason behind such solutions. I am trying to learn the underlying mathematical logic as to why some of these solutions work, as opposed to just memorizing a bunch of steps, without knowing why I am doing them; that's not learning, but rather just memorizing. Think of the old pedagogical analogy of teaching a man to fish, versus just giving him a fish. (Actually, even that's a poor analogy, I guess, since it might be better stated as, 'Teach a man how to fish, including why each step is necessary, rather than giving him a fish,' lol!)

As an example of a general rule that I am talking about, I get that you can't have zero in the denominator, as doing so results in an undefined situation, so I can easily see and understand how to handle that, no problem. What I am seeking is if there are other such rules that might exist which might be instrumental in dealing with limits, and why they are so.

Thanks again for your help.
 
I would say that, in effect, you skipped the problem, by not thinking carefully about it. If you didn't actually do the substitution (with care) and see whether it really did result in zero, then you didn't really do the problem.

I often tell students that the way to avoid errors in math is: (1) think; (2) write what you thought; (3) think about what you wrote; and (4) fix it. If you just imagine doing a problem without writing down the work, you can't see what is really happening, and you can't check that what you did makes sense. I really think that is the main principle here.

As far as principles go, specific examples are the best way to get to them, quite often. When I want to teach something general, I often start with an illustrative example. And if you think I'd take an example you offered and just do the work for you, you haven't spent enough time here! What I'd like you to do is to give an example and tell us what you thought about it, so we can discuss that. We don't give out free fish here! We want to watch you fish, and then tell you what's going wrong.

Your example is not a general rule at all; it's a specific rule. And if you have such "rules" for which you don't have a reason, that would be a great question to ask!
 
I do think that after some practice you can just look at SOME limit and state the answer. But you need to have the correct skills. The one that you stated is as flawed as can be. If I understand your logic, you are saying that if x is approaching infinity of some some function that x in the denominator then the answer is 0. That, as you found out, is not true.

A quick example is if you had x^2/x which equals x (if x is not 0). Now if x is approaching infinity than that limit will be infinity.

If the function instead was kx/x = k, then the limit would be k.
 
Here is a video that the recipe for limits as x goes to infinity but only for the division of two polynomials.
 
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