**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:

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

Very simple, everyone understands, everyone’s happy. What else does this mean, however? It means that the point 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 *x *= 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 *f *is headed straight for the point (3,11), and that’s what is meant by a limit.

**A 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

*f*does reach the value of 11 when

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

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 *x *= 4 from the right, you’ll notice that you approach a height of 2 when you get close to *x *= 4. This is, as you may have guessed, the right-hand limit for *x *= 4, and it is written as follows:

A left-hand limit is the height a function intends to reach as you approach the given *x *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

**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:

if the domain of *f* contains points arbitrary close to but smaller than *a* and *f*(*x*) approaches arbitrarily close to *L* as *x *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:

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

**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:

if the domain of *f* contains points arbitrary close to but greater than *a* and *f*(*x*) approaches arbitrarily close to *L* as *x *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:

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

**Limits and One-Sided Limits**

Keep in mind that for the (two sided limit of at , denoted

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

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

both one-sided limits exist and are equal 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.