Differential and Integral Calculus

One-sided limits.

Change the definition so that \(f(x) \) is only required to approach \(L\) when \(x \to a \) if \(x \) is greater than \(a \text{.}\) We say \(x \) "approaches \(a \) from the right," thinking of a number line. If the value of \(f(x) \) approaches \(L \) when \(x \) approaches \(a \) from the right, we say that the limit from the right of \(f(x) \) at \(x=a \) is \(L \text{,}\) and denote this \(\lim_{x \to a^+} f(x) = L \text{.}\) If we require \(f(x) \) to approach \(L \) when \(x \) approaches \(a \) but only for those \(x \) that are less than \(a \text{,}\) this is called having a limit from the left and is denoted \(\lim_{x \to a^-} f(x) = L \text{.}\)

Remark 4.6.

Just like wind directions (North wind, South wind, etc.), one-sided limits are named for the direction they come from, not the direction \(x \) is moving. Thus, \(\lim_{x \to 0^+} \) is evaluated by letting \(x \) approch zero from the positive direction, as shown to the right.

Both kinds of one-sided limits require something less stringent, so the statement \(\lim_{x \to a} f(x) = L \) automatically implies both \(\lim_{x \to a^+} f(x) = L \) and \(\lim_{x \to a^-} f(x) = L \text{.}\) Likewise, if \(f(x) \) is forced to approach \(L \) when \(x \) approaches \(a\) from the right, but also when \(x \) approaches \(a \) from the left, then this covers all \(x \text{,}\) and the (unrestricted) limit will be \(L \text{.}\) If you want, you can summarize this as a theorem -- wait, no it's too puny, let's make it a proposition. We won't be referring to this too often, but here it is.

Example 4.8.

Let \(f(x) = \lfloor x \rfloor \text{,}\) the greatest integer function. Let's evaluate the one-sided limits and two-sided limit at a couple of values. First, take \(a = \pi \text{,}\) you know, the irrational number beginning \(3.14\ldots \text{.}\) If we just look near this value, say between \(3.1 \) and \(3.2 \text{,}\) it is completely flat: a constant function, taking the value~3 everywhere. So of course the limit at \(x=\pi \) will also be~3. This is the same by words or pictures; see Figure 4.9.

By the formal definition, no matter what \(\varepsilon \) is chosen, you can take \(\delta = 0.1 \text{,}\) say, and \(f(x) \) will be within \(\varepsilon \) of~3 because it will be exactly~3. So the limit is~3, hence so are both one-sided limits as in the picture just above.

Now take \(x \) to be an integer, say \(a=5 \text{.}\) The limit from the right looks like it did before, with \(f(x) \) taking the value~5 for every sufficiently close \(x \) (here sufficient means within~1) greater than~5. On the other hand, when \(x \) is close to~5 but less than~5, we will have \(f(x) = 4 \text{,}\) as in the picture below. Thus,

\begin{gather*} \lim_{x \to 5^+} f(x) = 5 \\ \lim_{x \to 5^-} f(x) = 4 \\ \lim_{x \to 5} f(x) \text{DNE} \, . \end{gather*}

The two-sided limit does not exist because the two one-sided limits are not equal; see Figure 4.10.

Limits at infinity.

You have already seen the pictorial and verbal version of a limit at infinity. Here is the formal definition. It repeats a lot of the definition of a limit at \(x=a \text{.}\) The only difference is that instead of having to come up with an interval \([a-\delta , a+\delta]\) guaranteeing \(f(x) \) is within \(\varepsilon \) of the limit, you have to come up with an "interval near infinity". This turns out to mean an interval \([M,\infty) \text{.}\) In other words, there must be a real number \(M \) guaranteeing \(f(x) \) is with \(\varepsilon \) of \(L \) when \(x \gt M \text{.}\)

If a real number \(L \) exists satisfying this, we write \(\lim_{x \to \infty} f(x) = L \text{.}\) Sometimes to be completely unambiguous, we put in a plus sign: \(\lim_{x \to +\infty} f(x) = L \text{.}\)

Limits at \(-\infty \) are defined exactly the same except for a single inequality that is reversed. Now the implication that must hold is that for some (possibly very negative) \(M \text{,}\)

When this holds, we write \(\lim_{x \to -\infty} f(x) = L \text{.}\) When no such \(L \) exists, we write \(\lim_{x \to -\infty} f(x) \) DNE.

The graph of this function is shown in Figure 4.14. It has horizontal asymptotes at 1 and \(-1 \text{.}\) This suggests how to define a horizontal asymptote.

"Limits" of infinity.

Consider the function \(f(x) = 1/x^2 \text{,}\) defined for all real numbers except zero. What happens to \(f(x) \) as \(x \to 0\text{?}\) By our definitions, \(\lim_{x \to 0} 1/x^2 \) does not exist. But we can see that \(f(x) \) "goes to infinity". Because infinity is not a number, the limit technically does not exist. However, it is useful to classify DNE limits as ones where the function approaches \(\infty \) (or \(-\infty\)) versus ones where there is no consistent behavior.

Formally, this turns into the following defintion.

Again, if we reverse the last inquality to require that \(f(x) \lt N\) (and \(N \) can be a very negative number) we get the definition for a limit of negative infinity. Please remember these are all subcases of limits that don't exist! If you show that a limit is infinity, you have shown that the limit does not exist (and you have specified a particular reason it doesn't exist).

For one-sided limits and limits at infinity, the DNE case also includes a case where the limit would be said to be infinity. Stating all these would be repetitive. Try one, to make sure you agree it's straightforward.

Limit of a sequence.

A special case of limits at infinity is when the domain of \(f\) is the natural numbers. When \(f \) is only defined at the arguments \(1, 2, 3, \ldots \text{,}\) it is more usual to think of it as a sequence \(b_1, b_2, b_3, \ldots \text{,}\) where \(b_k := f(k) \text{.}\) The definition of a limit at infinity can be applied directly, resulting in the definition of the limit of a sequence.

Pictorially, if a sequence has a limit \(L \text{,}\) then for every pair of parallel horizontal lines, however narrow, enclosing the height \(L \text{,}\) the sequence must eventually stay between them. This is shown in Figure 4.21.

As you will see, Proposition 4.27 and Proposition 4.28 give ways to determine limits of more complicated functions once you understand limits of some basic functions. Here is another piece of logic that can help do the same thing. You will prove it in your homework.

The same fact is true of sequences: if \(a_n \leq b_n \leq c_n\) for these three sequences and the first and last sequence converge to the same limit \(L \text{,}\) then so does the middle one. We will not do anything with this now, but will get back to this fact in a week or two. The next exercise brushes up on the logical syntax of limits.

Continuity on regions.

Before going on to use the notion of continuity to help us compute limits, we will state one famous result which will seem either stupid and obvious or deep and tricky.

This says, basically, a continuous function can't get from one value to another without hitting everything in between. The theorem is most often used when there is a number we can only define this way. For example, let \(f(x) := e^x / x \text{,}\) which is an increasing function on the half-line \([1,\infty) \text{.}\) We want to say "let \(c \) be the value for which \(f(c) = 3 \text{.}\)" How do we know there is one? Well, \(f(1) = e \text{,}\) which is less than 3, and \(f(3) \approx 6.695 \) which is greater than 3. So there must be an argument between 1 and 3 where \(f \) takes value 3. There can be only one because \(f \) is strictly increasing (you can prove after another two sections).

Some more techniques and tricks.

This course is more about using limits than it is about computational technique, but you should at least see some of the standard techniques for cases that go beyond what's in Proposition 4.27 and Proposition 4.28.

Suppose you need to evaluate \(\lim_{x \to a} f(x) / g(x) \text{.}\) If both \(f \) and \(g \) have nonzero limits at \(a \text{,}\) say \(L \) and \(M \text{,}\) then Proposition 4.28 tells you

In fact if \(L=0 \) but \(M \neq 0 \text{,}\) this still works. If \(M=0 \) but \(L\neq 0 \text{,}\) then the question of evaluating \(\lim_{x \to a} f(x) / g(x) \) also has an easy answer.

The remaining case, when \(L = M = 0 \text{,}\) can be enigmatic. Calculus provides one solution you will see in a few weeks (L'Hôpital's Rule), but you can often solve this with algebra. If you can factor out \((x-a) \) from both \(f \) and \(g \text{,}\) you may get a simpler expression for which at least one of the functions has a nonzero limit.

Sometimes you have to do a little algebra to simplify. Here's an example of one of the most common simplification tricks.

This algebra trick occurs so commonly throughout mathematics that you should always think about conjugate radicals every time you see an expression with a square root added to or subtracted from something!

Further tricks can wait until you've learned some more background. Although limits are needed to define derivatives, you can then use derivatives to evaluate more limits (L'Hôpital's Rule). Similarly, limits are used to define orders of growth, which can then be used to evaluate more limits.