Limit of a Function: Survey of Methods

If you wish to simultaneously follow another text on limits of functions in a separate window, click here for Theory and here for Solved Problems.

The limit is one of the crucial tools in investigating functions. 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 limit problem 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 function at a given point might have a limit which is a real number (a proper limit). In this case we say that the function converges (or is convergent) at that point. Otherwise the function is divergent there. We use the same classification for "the problem of finding the limit at the point", which we call "limit" for short. So we would say that the given limit (a problem to solve) is convergent (there is a proper limit - a result) or divergent.
Among divergent limits (i.e. problems) 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 limits 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?

We start by specifying the most important situation:

Question:
Let a function f be defined on some reduced neighborhood of a point a by a certain algebraic formula. Find its limit at a.

Alternative questions:
Let a function f be defined on some reduced right neighborhood of a point a by a certain algebraic formula. Find its limit at a from the right.
Let a function f be defined on some reduced left neighborhood of a point a by a certain algebraic formula. Find its limit at a from the left.

These questions are handled in the same way, see below.

Before showing what to do we briefly look at other situations. What if the function is not given by one common formula on some reduced (one-sided for one-sided limits) neighborhood of a? The good case is when the function is given by one formula on the right and another formula on the left of a and we want the two-sided limit. Then we can pass to one-sided limits, where we are in the situation as described in the Question, so we use the algorithm outlined below, find the answers (if we are lucky) and compare. If the function is so weird that it is not even given a by a nice formula on a one-sided neighborhood of a, then we have to work the problem out individually drawing on our understanding of limits, there is no algorithm.

Now back to the Question.

Solution:
Step 1. "Plug in" the point a into the given expression and try to find the outcome using the limit algebra. In particular one needs to know how to substitute infinity into elementary functions. It might be necessary to work with one-sided limits and/or one-sided results of limits, see Note 1 below, this is most often the case when you encounter the expression 1/0 (see appropriate box below).

Sometimes it helps to simplify the expression before substituting a. This is especially handy when substituting infinity to powers with negative exponents, many people like them better in the form of fractions.

What can happen?

a) If the limit algebra gave a definite answer (a number, infinity or minus infinity, or that the limit DNE, see Note 2 below), then this answer is also the answer to the limit problem and you are done. Note that the algebra of limits, especially when featuring some infinities, is not a "real algebra", so it might be better not to write it as a part of the "official solution".
Example.

b) The other possibility is that the limit algebra did not lead to an answer because something went wrong. In that case you have to try some trick, that is, you go to Step 2.

Warning! Sometimes you are tempted to substitute only to "nice" parts of the expression and leave the rest for later. This does not work in general! The only way to do limits "by parts" is by splitting them into more limits, see this note. For one limit, you either substitute everywhere or not at all.

Step 2. If the limit algebra 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 a in Step 1, although it failed, should still do a very valuable service, namely it should identify what kind of problem you have. This will 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 at infinity",
  • box "polynomials at proper points" (cancelling),
  • box "difference of roots".

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:
  • box "comparison and oscillation", which typically includes limits like the limit of sin(x) at infinity.

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 "a nice outer function",
  • box "substitution".

As a bonus we add
  • box "equivalent infinitesimals".

Sometimes the method from the appropriate box will get you the answer. But 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 or until you give up.

It is important to keep an eye open for simplifications. In particular, before applying tricks from some box, you should check whether you need to apply this trick to the whole given expression. Sometimes the trouble is caused just by a part of the expression and the rest is "nice", then it is usually very wise to split the given limit into several parts and apply the most convenient method to each part. Sometimes it is the other way around, you are forced to split a limit since the trick you want only applies to a part of the given limit. For a more detailed discussion on splitting limits and evaluating parts of it, see this note.

Likewise, in case you have to go back to Step 1, it is strongly recommended that you look at what you got after applying tricks and simplify if possible.

Note that there are limit problems that do not fit any pattern we covered here (that is, they do not exactly fit any box below). 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.

Beginners have sometimes trouble with the proper notation of their calculations and result.

Note 1.

One-sided results of limits are often needed when substituting into functions. This information about the result is usually obtained from the given limit itself (when it is given one-sided), but sometimes it also follows from the nature of the problem.

Example: The limit of ex − 1 for x→0+ is 0+.
Indeed, substituting x = 0 into the expression we get 0, now we need to find out what kind. If x→0+, then x is a number close to 0 that satisfies x > 0. Then also ex > 1, therefore ex − 1 > 0.

Example: The limit of 1 + 2x − x2 for x→2- is 1+.
Indeed, substituting x = 2 into the expression we get 1, now we need to find out what kind. If x→2-, then x is a number close to 2 that satisfies x < 2. However, it is not clear what happens to the expression. There are two ways to find out.

Mathematically correct is this: 1 + 2x − x2 = 1 + x⋅(2 − x). If x < 2, then (2 − x) > 0, and x is positive when close to 2, therefore x⋅(2 − x) > 0 and 1 + x⋅(2 − x) > 1, which proves our claim.

This method was correct but specific for this particular example. In general one can always try to investigate monotonicity of the given expression around the limit point, but in practice people rarely bother to spend the time. Usually they use a different method, which is much more convenient, but it has the small disadvantage of not being mathematically correct. Here it comes:

If x→2-, then x is something like 2 minus something very very tiny. Say, x = 1.9999. When we substitute this into the expression 1 + 2x − x2, we get 0.99980001 < 1, which suggests that we approach the limit result 1 from below. Of course, this does not prove anything, for some wild functions this method would fail. However, most problems feature "reasonable" functions, and given how simple this method is, most people (including me) use it quite frequently when faced with a more complicated problem of "one-sidedness".

Example: The limit of 1 + cos(x) for x→0 is 2-.
Indeed, substituting x = 0 into the expression we get 2. Moreover, if x is a number close to 0 (no matter on which side), then cos(x) < 1 (note that "close to 0" does not include 0 itself, so the sharp inequality is true). Therefore 1 + cos(x) < 2, proving the claim.

Example: The limit of 1 + sin(x) for x→0 is just 1.
Indeed, substituting x = 0 into the expression we get 1. However, we cannot claim that it would be 1+ or 1-. As x→0, it is sometimes negative and sometimes positive (it is a two-sided limit), hence also sin(x) is sometimes positive and sometimes negative. No matter how small reduced neighborhood of 0 we consider, the function 1 + sin(x) is both greater and smaller than 1 on it, thus no one-sided conclusion is possible.

Another nice example with one-sidedness is in this note.

Note 2.

We know fairly well how to recognize cases when the limit exists (the limit algebra) and when it is troublesome (the indeterminate expressions). Less studied, but equally important is the ability to recognize which limits do not exist. This is usually ignored in calculus courses, since after all we are expected to solve the problems in school.

Still, for the sake of completness, we drop a word or two about limits that do not exist. A good start is to remember some "famous" limits that do not exist. The most frequent examples are the limits of the sine and the cosine at (minus) infinity and the ratio 1/0 with 0 not being one-sided, of note are also limits of the cotangent at 0 and its shifts and tangent at π/2 and its shifts.

Then it helps to know how such problematic expressions interact with other expressions. There are many possible situations, we tried to collect the more interesting ones in an attempt at the algebra of N (N for non-existent), where we also included indeterminate expressions.

This topic is so ignored that our list is probably the only one you are going to see anywhere, which suggests that perhaps this is not so important after all.