On this screen, we’re going to look quickly at some functions where the limit exists at a particular point, and then consider some where the limit does *not* exist.

Reminder: \[\text{The limit of the function $f(x)$ as $x$ approaches $a$, written} \] \[\lim_{x \to a} f(x) = L \] \[\text{is a number $L$ (if one exists)}\] \[\text{such that $f(x)$ is $\underbrace{\text{as close to}}_{\Large{\text{within } \pm

\epsilon} \text{ of}}$ $L$ as we want whenever $x$ is $\underbrace{\text{sufficiently close to}}_{\Large{\text{within }

\pm \delta \text{ of}}}$ $a.$ }\]

We’re going to look at several functions where the limit *does* exist, with an emphasis on how we know the limit exists. While the result that “the limit exists” for each of these functions may seem obvious and trivial, we need to lay this groundwork in order to recognize when a limit does not exist, both further down on this screen, and in your homework and exam questions.

Let’s first consider $\displaystyle{\lim_{x \to 4}\left[ 6 – (x-5)^2 \right] = 5}.$ (There’s nothing special about the point $x=4;$ we could have chosen any point of interest.)

Graph of $f(x) = 6 – (x-5)^2$ versus *x*,

showing $\displaystyle{ \lim_{x \to 4}f(x) = L=5}$

showing $\displaystyle{ \lim_{x \to 4}f(x) = L=5}$

The graph of this function is the upside-down parabola shown. We’re focused on the limit as $x \to 4$: as you can see, you can get as close to the limit value $L=5$ as you’d like by being sufficiently close to $x=4.$ In fact in this case the function is defined at $x=4,$ with $f(4) = L,$ but that’s *irrelevant*: focusing on the *limit,* we care only that we can get as close to $f(x)=L$ as we’d like by being sufficiently close to $x=4.$

Let’s examine another function at a different point of interest.

Consider $\displaystyle{\lim_{x \to 0}\dfrac{\sin(x)}{x} = 1.}$

Graph of $g(x) = \dfrac{\sin(x)}{x}$ versus *x*,

showing $\displaystyle{ \lim_{x \to 0}g(x) = L=1}$

showing $\displaystyle{ \lim_{x \to 0}g(x) = L=1}$

The function $g(x) = \dfrac{\sin(x)}{x}$ is *not* defined for $x=0$, but you can see visually that $L=1$ … meaning, of course, that you can get as close to $L=1$ as you’d like by being sufficiently close to $x=0,$ as long as you’re not actually *at* $x=0.$

Let’s consider another function at yet another point of interest.

Consider now $\displaystyle{\lim_{x \to 3}\dfrac{\sqrt{x+1}-2}{x-3} = L.}$

Graph of $h(x) = \dfrac{\sqrt{x+1}-2}{x-3}$ versus *x*,

showing $\displaystyle{ \lim_{x \to 3}h(x) = L}$

showing $\displaystyle{ \lim_{x \to 3}h(x) = L}$

This function is not defined for $x=3,$ but we can see that the limit exists, even if we don’t know immediately what its value is (although we can see visually it is around 0.25). The fact that we don’t know the value is of no consequence currently: we care only that the limit *exists,* since we can get as close to the value *L* as we’d like by being sufficiently close to $x=3.$ (We could, if we cared to, certainly approximate its value to within whatever error bounds we choose, by using the technique of the lab on the preceding screen.)

When a limit exists, you can point to its location

Some Limits that Exist, Summary The key thing to notice in each cases above is that

DNE = “Does Not Exist”

Let’s now introduce some cases where the limit is does Consider this function that “jumps” in value at $x=0:$

\[t(x) =

\begin{cases}

-3 & \text{for } x \lt 0 \\

\phantom{-}5 & \text{for } x \ge 0

\end{cases}\]

Graph of *t(x)* versus *x*,

showing $\displaystyle{ \lim_{x \to 0}t(x)}$ does*not* exist

showing $\displaystyle{ \lim_{x \to 0}t(x)}$ does

This function “jumps” at $x=0$ from $-3$ to 5, and hence the limit at $x=0$ does not exist: there is no single number *L* that we can be as close to as we’d like for values of *x* both less than *and* greater than $x=0.$ That is, there is no single place on the graph you can point to that represents the limit at $x=0.$

Notice that at all *other* points the limit *does* exist. For example, considering just two particular points: $\displaystyle{\lim_{x \to -4}t(x) = -3},$ and $\displaystyle{\lim_{x \to 2}t(x) = 5}.$

But at $x=0$:

\[\lim_{x \to 0}t(x) = \text{DNE} \]

Graph of $j(x) = \sin(1/x)$ versus *x*,

showing $\displaystyle{ \lim_{x \to 0}j(x)}$ does*not* exist

showing $\displaystyle{ \lim_{x \to 0}j(x)}$ does

*Tip*: On a mobile device, to zoom only horizontally place your fingers near the *x-*axis and then “stretch” it. On desktop or laptop, place your cursor near the *x-*axis and hold the **Shift-key** while you scroll.

Let’s first think this through as if we didn’t have the graph to rely on, since on an exam you might not be provided with one or the means to create one.

This function is not defined for $x=0,$ which in and of itself tells us *nothing* about the limit there. Instead, to think about the limit we must think about what happens as *x* takes on values closer and closer to 0. We can reason as follows:

First, the inner function, $\dfrac{1}{x},$ becomes larger and larger in size, growing and growing and growing (1) in the positive direction if we’re on the right side of 0, so for $x \gt 0,$ and (2) in the negative direction if we’re on the left side of 0, for $x \lt 0.$ The key point is that $1/x$ never approaches a single value as $x \to 0$; instead, its value keeps changing, growing and growing in size (in either the positive or negative direction).

Second, recall that the sine function oscillates between $-1$ and 1:

\[-1 \le \sin(\text{whatever}) \le 1 \quad \text{Always.}\]
That means that as $1/x$ grows and grows (either positively or negatively) as $x \to 0$, always changing values, the sine function keeps oscillating between $-1$ and 1. As it turns out, $\sin (1/x)$ oscillates faster and faster the closer we get to $x=0,$ as you can see in the graph above. No matter how close you get to $x=0,$ there are some values of *x* with function output values equal to 1, other values of *x* with function output values equal to $-1,$ and all other values lie between those two extremes.

Hence there is *no single value L* for the function’s output value that we can get as close to as we’d like by being sufficiently close to $x=0,$ and the limit does not exist (DNE):

\[ \lim_{x \to 0} \sin(1/x) = \text{ DNE} \]

Of course we can reach this conclusion much more quickly by relying on the Desmos graph: if you move the blue dot near $x=0$ from both the left and the right sides, you will see that there is no single value *L* we can be as close to as we’d like by being sufficiently close to $x=0.$ Instead, as we move closer and closer, the values continue to oscillate around — and hence the limit DNE.

Graph of $r(x) = \dfrac{1}{(x-2)^4}$ versus *x*,

showing $\displaystyle{ \lim_{x \to 2}r(x)}$ does*not* exist

showing $\displaystyle{ \lim_{x \to 2}r(x)}$ does

The function $r(x) = \dfrac{1}{(x-2)^4}$ is not defined for $x=2,$ which – once again – in and of itself tells us nothing about the limit there. But, we see that as we get closer and closer to $x=2,$ the values of the function become larger, and larger, and larger still . . . without bound. **There is no single place you can point to on the graph that we can get as close to as you’d like, and hence the limit does not exist.**

“infinity” means

“arbitrarily large”

You may be thinking that “the limit as $x=2$ is infinity,” and if so you are correct. But we must be very clear: “arbitrarily large”

So both of these statements are true:

- $\displaystyle{ \lim_{x \to 2}r(x)}$ does not exist (DNE) because there is no single number we can get as close to as we’d like by being sufficiently close to $x=2$: \[\lim_{x \to 2}\dfrac{1}{(x-2)^4} = \text{DNE}\]
- This limit does not exist in a particular way, because the function grows without bound and can be made arbitrarily large near $x=2.$ We can convey this additional information by writing

\[\lim_{x \to 2}\dfrac{1}{(x-2)^4} = \infty\] This is a valid mathematical statement that you will see, and you will probably write similar statements yourself. We emphasize again that the statement indicates merely that the limit does not exist*because*the function just grows and Grows and GROWS. The statement does*not*mean that there is a magical place called “infinity” that we can put our finger on and get as close to as we’d like.

When you see the limit for a function like this, you’re like to encounter both $\displaystyle{ \lim_{x \to 2}r(x)} = \text{DNE}$ and $\displaystyle{ \lim_{x \to 2}r(x)} = \infty$ because both are correct. The second simply conveys more information than the first, specifying the

Some Limits that Do Not Exist: Summary

We’ve seen three ways in which a limit does not exist at $x=a$:

- the function “jumps” from one output value to another at $x=a$;
- the function oscillates between $-1$ and 1 no matter how close you get to $x=a$;
- the function “blows up” — grows without bound — as you approach $x=a.$ In this case, if we choose we can provide more information by specifying $\displaystyle{\lim_{x \to a}f(x) = \infty }$ (or, as we’ll see, in other cases $-\infty$.)

Let’s consider a few functions for which students often have conflicting thoughts regarding whether the limit exists or not.

Question 1: Constant Function

Two students are discussing a problem involving the constant function $c(x) = 95$ and its graph.

They’ve been asked to find the limit as $x \to 5: \; \displaystyle{\lim_{x \to 5} c(x) = \, ? }$

JAMES: The limit must be 95, because what else can it be?

MICHAEL: But that’s weird. The function just *sits there* at 95! As $x \to 5$ [Michael gestures toward $x=5$ on the graph], the line doesn’t come up toward that value from underneath, or come down toward it from above. So it makes no sense to talk about the limit. It just doesn’t have one, so I think the limit does not exist. DNE!

Question 1 illustrates an important point: We mentioned earlier that mathematicians took hundreds of years to settle on the definition of “limit” that we now use. Part of the challenge of developing a usable definition is that it must be applicable to *any* function, and return a result that is useful in our larger framework. Indeed, it would be problematic (as we’ll see later) if Michael were correct and this limit did not exist: we would have a whole separate way of dealing with constant functions instead of the framework we’ll use for every other function. Instead, the definition was developed to return a useful value (the only possible value) for constant functions, just as it does for many other situations we will examine.

Let’s consider another problem.

Question 2: $\displaystyle{\lim_{x \to 0} x \cos(1/x) }$

Two students are discussing a problem involving the function $f(x) = x \cos(1/x).$ Its interactive graph is below. They’ve been asked to find the limit as $x \to 0: \; \displaystyle{\lim_{x \to 0}x \cos(1/x) = \, ?}$

Graph of $f(x) = x \cos(1/x)$ versus *x*

LUIS: Looks to me like the limit is zero because that’s what the function is tending toward as you approach $x=0$ from either side: $\displaystyle{\lim_{x \to 0}x \cos(1/x) = 0.}$

QIAO: Yes, but I think they’re trying to trick us! Remember that function above, $j(x) = \sin(1/x)?$ That limit doesn’t exist because the function keeps oscillating and going up and down and up and down no matter how close to $x=0$ you get. This one does the same thing, so this limit doesn’t exist either: $\displaystyle{\lim_{x \to 0}x \cos(1/x) = \text{ DNE}.}$

We’ll examine two of the ways limits don’t exist on the upcoming screens, starting on the next screen with the function above that “jumps” from one value to another.

For now, what questions or comments do you have about the material on this screen? Please let your fellow community members, including us, know over on the Forum!

- The limit of a function at $x=a$ exists and equals
*L*when we can get as close to*L*as we’d like by being sufficiently close to $x=a.$ When the limit exists, we can literally point to its height at $x=a$ on a graph of the function. - If there is no single value
*L*we can be as close to as we’d like by being sufficiently close to $x=a,$ then the limit does not exist, often abbreviated “DNE.” We saw three ways in which a function does not exist:- the function “jumps” from one value to another at $x=a;$
- the function continues to oscillate between two values, say $-1$ and 1, as you approach $x=a;$
- the function grows without bound as $x \to a.$ In this case we may write that the limit equals $\infty$ (or $-\infty$); we’ll examine this case in more detail in a few screens.