One-Sided Limits

One-Sided Limits

What Is a Limit?

When I first took calculus in high school, I was hip-deep in evaluating limits via tons of different techniques before I realized that I had no idea what exactly I was doing, or why. I am one of those people who needs some sort of universal understanding in a math class, some sort of framework to visualize why I am undertaking the process at hand. Unfortunately, calculus teachers are notorious for explaining how to complete a problem (outlining the steps and rules) but not explaining what the problem means. So, for your benefit and mine, we’ll discuss what a limit actually is before we get too nutty with the math part of things.

Let’s start with a simple function:

f(x)=2x+5

You know that this is a line with slope 2 and y-intercept 5. If you plug x = 3 into the function, the output will be

f(3)=2.3+5=11

Very simple, everyone understands, everyone’s happy. What else does this mean, however? It means that the point (3,11) belongs to the relation and function I call f. Furthermore, it means that the point (3,11) falls on the graph of f(x), as evidenced in Figure.

The point (3,11) falls on the graph of f.

All of this seems pretty obvious, but let’s change the way we talk just a little to prepare for limits. Notice that as you get closer and closer to x = 3, the height of the graph gets closer and closer to y = 11. In fact, if you plug x = 2.9 into f(x), you get f(2.9) = 2(2.9) + 5 = 10.8.

If you plug in = 2.95, the output is 10.9. Inputs close to 3 give outputs close to 11, and the closer the input is to 3, the closer the output is to 11.

Even if you didn’t know that f(3) = 11 (say for some reason you were forbidden by your evil step-godmother, as was Cinderella), you could still figure out what it would probably be by plugging in an insanely close number like 2.99999. I’ll save you the grunt work and tell you that f(2.99999) = 10.99998. It’s pretty obvious that is headed straight for the point (3,11), and that’s what is meant by a limit.

limit is the intended height of a function at a given value of x, whether or not the function actually reaches that height at the given x. In the case of f, you know that does reach the value of 11 when = 3, but that doesn’t have to be the case for a limit to exist. Remember that a limit is the height a function intends to reach.

One-sided Limits

Occasionally, a function will intend to reach two different heights at a given x, one height as you come from the left side, and one height as you come from the right side. We can still describe these one-sided intended heights, using left-hand and right-hand limits. To better understand this bizarre function behavior, look at the graph of h(x) in Figure .

The graph of h(x) consists of both pieces; a graph like this is usually the result of a piecewise defined function.

This graph does something very wacky at = 4: it breaks. Trace your finger along the graph as it approaches = 4 from the left. What height is your finger approaching as you get close to (but don’t necessarily reach) = 4? You are approaching a height of 6. This is called the left-hand limit and is written like this:

\lim_{x\rightarrow 4-}h(x)=6

The little negative sign after 4  indicates that you should only be interested in the height the graph approaches as you travel along the graph from the left-hand side. If you trace your finger along the other portion of the graph, this time toward = 4 from the right, you’ll notice that you approach a height of 2 when you get close to = 4. This is, as you may have guessed, the right-hand limit for = 4, and it is written as follows:

\lim_{x\rightarrow 4+}h(x)=2

A left-hand limit is the height a function intends to reach as you approach the given value from the left; the right-hand limit is the intended height as you approach from the right.

Until now, we have only spoken of a general limit (in other words, a limit that doesn’t involve a direction, like from the right or left). Most of the time in calculus, you will worry about general limits, but in order for general limits to exist, right- and left-hand limits must also be present; this we learn in the next section, which will tie together everything we’ve discussed so far about limits.

Can you feel the electricity in the air?

Fig 1.1

One sided limits

\lim_{x\rightarrow a}f(x)=L=\lim_{x\rightarrow a+}f(x)=\lim_{x\rightarrow a-}f(x)

\lim_{x\rightarrow b-}f(x)=M

\lim_{x\rightarrow b+}f(x)=N

Left-Hand Limits : We say that a function f has the left-hand limit M as x approaches b, or that f approaches M as x approaches b from the left, and we write:

\lim_{x\rightarrow b-}f(x)=M

if the domain of f contains points arbitrary close to but smaller than a and f(x) approaches arbitrarily close to L as approaches a from the left (x increases toward a).

In Fig. 1.1, f  has a left-hand limit of L at x = a and also a left-hand-limit of M at x = b, so that:

\lim_{x\rightarrow a-}f(x)=L

\lim_{x\rightarrow b-}f(x)=M

The formal definition of left-hand limits is stated as follows. A function f has a left-hand limit L as x approaches a and we write

\lim_{x\rightarrow a-}f(x)=L

Right-Hand Limits : We say that a function f has the right-hand limit N as x approaches b, or that f approaches N as x approaches b from the right, and we write:

\lim_{x\rightarrow b+}f(x)=N

if the domain of f contains points arbitrary close to but greater than a and f(x) approaches arbitrarily close to L as approaches a from the right (x decreases toward a).    In Fig. 1.1, f  has a right-hand limit of L at x = a and also a right-hand-limit of N at x = b, so that:

\lim_{x\rightarrow a+}f(x)=L

\lim_{x\rightarrow b+}f(x)=N

The formal definition of right-hand limits is stated as follows. A function f has a right-hand limit L as x approaches a and we write

\lim_{x\rightarrow a+}f(x)=L

Limits and One-Sided Limits

Keep in mind that for the (two sided limit of f(x) at x=a, denoted

\lim_{x\rightarrow a}f(x)

Both sides of a must be considered. It’s clear that

limit exists \Rightarrow both one-sided limits exist and are equal to it;

both one-sided limits exist and are equal \Rightarrow limit exists and is equal to them

Note that:

  • If both one-sided limits exist but are different, then the limit doesn’t exist.
  • If one or both of the one-sided limits do not exist, then the limit doesn’t exist.
  • If  a limit doesn’t exist there is no conclusion about the one-sided limit except this fact: It’s not true that both exist and are equal.It may be that one or both of them do not exist. It may be that both exist but are different.