Limit of a Sequence: Survey of Methods

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The question of convergence is the central question in investigation of a sequence. As with any "good" question, finding an answer is not always easy and often one has to overcome problems. For many of them we have specific methods and tricks. If you want to be proficient in evaluating limits, it is important to develop (by practicing) mental "boxes" of problems, each box holding limit problems of a specific type. When you encounter a problem, you find the corresponding box and pull out the corresponding method of solution. Often one does not get an answer in this way, just the sequence changes into a different one, so several tricks have to be used one after another. There are also problems which do not fit into any box, then the only hope is experience and intuition.

First we briefly review what kind of answer one can get. A given sequence might have a limit which is a real number (a "proper limit"). In this case we say that the sequence "converges" (or "is convergent"). Otherwise the sequence is "divergent".
Among divergent sequences some are "nicer": they tend to infinity or to minus infinity, that is, they have an "improper limit". If there is some limit - finite for convergent sequences or infinite - we say that the limit "exists", since we still get some information. The last case is when there is no limit at all, finite nor infinite. In this case we say that the limit does not exist, which is often shortened as DNE.

How to find limits?

Question: Evaluate the limit .

Solution: First we will do a brief overview of the solution procedure, then add more details in notes.

Step 1. "Plug in" infinity into the given expression an. Typically you would evaluate small and simple parts of the expression individually using your knowledge of elementary limits and then put the parts together using "limit algebra". In many problems it helps if you know and apply intuitive calculation, especially the scale of powers.

Sometimes it helps to simplify the expression before substituting infinity, one can avoid unnecessary trouble this way. This especially applies to powers with negative exponents, for instance it is better to replace n−1 with 1/n since the latter is more intuition-friendly.

What can happen?

a) Sometimes you get an answer right away. There are two possible kinds of a definite answer.

First, the limit algebra (and intuitive calculations) could lead to a definite expression (a real number, infinity, or negative infinity); this is the answer to the limit question then. Note that the algebra of limits and infinity is not a "real algebra", so one should not write it as a part of the "official solution".

Second, the limit algebra (and intuitive calculations) could lead to an expression that is known not to have a limit (for instance using "algebra of DNE"); then the answer is that the limit does not exist (see Note below).

b) The other possibility is that the "plugging in of infinity" did not lead to an answer because something went wrong. In that case you have to try some trick. That is, it is time to pass to the next step.

Step 2. If intuitive calculations and algebra of infinity failed, then there must have been some problem. For many kinds of problems there are quite reliable methods, so one should also know something about the more popular problems (for instance about indeterminate expressions). The substitution of infinity in Step 1, although it failed, should still do a very valuable service, namely it should identify what kind of problem you have. This should help you in fitting your problem into an appropriate "box", then you just apply the method recommended in this box.

Indeterminate expressions are the prevalent reason for failure in Step 1, and fortunately for each of them there is a special box with a suitable method.
  • box "1/0",
  • box "indeterminate ratio" , ,
  • box "indeterminate product" ∞⋅0,
  • box "indeterminate difference" ∞ − ∞,
  • box "indeterminate power" 1, 00, 0.

However, often it is better to skip these general boxes and instead use a box that specializes on a certain expression appearing in the limit:
  • box "polynomials, sums and ratios with powers",
  • box "difference of roots",
  • box "exponential", that is, and similar expressions,

Then there is a box that is not narrowly focused on a certain type of expression or a problem, but rather offers a more general method of dealing with oscillations and problems that are difficult to handle (e.g. when they include expressions that cannot be differentiated):
  • box "comparison and oscillation", which typically includes limits like

Finally there are two boxes with methods that do not solve anything by themselves, but they can sometimes get us much closer to this solution by significantly simplifying the given limit:
  • box "sequence in a nice function",
  • box "substitution".

Sometimes the method from the appropriate box will get you the answer. But quite often you get another limit to evaluate, which means that you should go back to Step 1 and start again and possibly again, until you get the answer. In case you have to go back to Step 1, simplifying the limit expression before going there can be crucial for the success of calculation. This is especially true if the expression is more complicated. In such cases one can often identify "nice" parts of the expression, whose limits are known and which are not parts of the "problem" we face; it is usually very smart to separate these "nice" parts using the theorem on limit and operations, and evaluate them separately. For examples and important notes see this note.

Note that there are sequences that do not fit any pattern we covered here (that is, they do not exactly fit any box above). Then the more experience and understanding of the concept of limit you have, the better your chances of evaluating the limit.

This outline should make more sense if you look at some Solved Problems - Limits and compare how they are solved with the general solution description here. You will also find some other useful tricks there.

Warning! Sometimes the sequence in your problem has some parts that you can figure out right away and some parts that will need further work. You may be tempted to simply substitute the answers for the "nice" parts and then work out the rest; that is, you may want to substitute infinity just for some n and leave the others for later. This does not work in general! For more details, see this note. In general, you either substitute everywhere or not at all.

Beginners have sometimes a related trouble when they are not sure how long to keep the "lim" symbol in calculations. For some insight see this note.

Notes on Step 1.

After you substitute infinity into a given sequence, you should be able to tell what happens. For getting "positive" answers, when you get a limit, you need to know elementary limits, the scale of powers and the "limit algebra". The knowledge of indeterminate expressions is crucial for recognizing that you have a problem that should be further investigated, it also allows you to pick the right box for it. All this is usually fairly well mastered by students.

Less studied, but equally important is the ability to recognize which limits do not exists. A good start is to remember that the following limits do not exist:

There is more of these, different ways in which limit may fail to exist can be found in the algebra of DNE.