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A.1 Limits Introduction

Let’s dive right in and examine a simple function so we can make the distinction between a function’s output-value at a given input-value, and its limit-value near that same input-value. We’ll of course also address errors students often make when first learning about limits.

An initial function to examine: $f(x) = \dfrac{x^2 \, -\, 4}{x\,-\,2}$

To begin our study of limits, let’s consider a single function: $f(x) = \dfrac{x^2 \, – \, 4}{x \, – \, 2},$ graphed here:

Graph of the function we're using for our limits introduction: f(x) = (x^2 - 4)/(x - 2), which is a line with a hole at the point (2, 4).

Why is the graph a line? Notice that we can factor the numerator, and so
\[\frac{x^2\, -\, 4}{x\,-\,2} = \frac{(x+2)\cancel{(x-2)}}{\cancel{(x-2)}} = x+2,\] which is a line with slope equal to 1 and y-intercept equal to 2.

Now, please consider this question:

Looking at the graph, what does $f(2)$ equal?

Before proceeding, ask yourself what thoughts are going through your head.

Many people find themselves feeling torn, or at least can imagine someone feeling torn. On the one hand, you know that $x=2$ is not in the function’s domain since $f(x) = \dfrac{x^2\, -\, 4}{x\,-\,2}$ is undefined for $x=2.$ And indeed, there is a hole in the function’s graph at the point where $x=2$ indicating that the function is undefined there.

On the other hand, a part of you may want to say “$f(2) = 4,$” essentially filling in that hole in your mind. There’s almost a pull to say something like, “$f(2)$ would equal four if the function were defined there.” But it’s not defined there. Hmmmm.

First, let’s be perfectly clear: $x=2$ is not in the domain of f, and so
\[f(2) \text{ is undefined.} \quad \cmark\] That’s the answer to our question above.

And yet there is more information to be had. We can say absolutely correctly, for instance,

“$f(x)$ is close to 4 when x is close to 2.”

Or we can make the preceding statement more precise:

“We can get as close to $f(x)$ = 4 as we’d like by being sufficiently close to x = 2.”

We capture both of these true mathematical statements by writing
\[\lim_{x \to 2}f(x) = 4\] which is read as

“The limit of $f(x)$ as x approaches 2 equals 4.”

And that is our first mathematically correct “limit statement.”

The limit says nothing about the value at x = 2.
Notice that this limit statement says nothing about the value of $f(x)$ at x = 2. As we saw above, $f(x)$ is undefined at that point, which is information not captured by the limit statement. Instead, the limit statement tells us about the function’s value when we are close to x = 2 — as close as we care to be, as long as we’re not actually at x = 2.

The interactive graph below allows you to explore these statements dynamically. You can change the x-value of the red dot (initially at x = 0.94), and the point’s label automatically displays the $y=f(x)$-value (initially $y = f(0.94) = 2.94$). Move the point toward x = 2, and zoom in, and then move the point closer still; move the point toward x = 2 from both the left and right sides. We know the results are obvious and may leave you feeling underwhelmed, but in this limits introduction we want (even need) you to gain a physical sense for what it means when you see the statement $\displaystyle{ \lim_{x \to 2}f(x) = 4 }$:

“We can get as close to $f(x)$ = 4 as we’d like by getting sufficiently close to x = 2.”

Interactive graph of $f(x) = \dfrac{x^2\, -\, 4}{x\,-\,2}$ versus x

You may tire of zooming-and-moving-closer. If not, you’ll probably find that even Desmos will only let you zoom in so far. Either way, continue to imagine the zooming-and-moving-closer process in your mind, so that even if we asked you to get to within, say, 0.000000000001 of $f(x) = 4,$ you could move the input x-value close enough to x = 2 to achieve that goal. (And you could do the same if we asked you to get within $10^{-21}$ of $f(x) = 4,$ or to within $10^{-37},$ or any tiny value we chose.)

Generalizing to a working definition of “limit”

This may all seem seem conceptually simple to you — if so, great. (And if not, no worries: we’re going to be developing increased understanding as we continue, so please keep going.) At the same time, it took mathematicians a tremendous amount of work and hundreds of years — truly, hundreds of years — to develop the precise formal definition of “limit” that we use today. We in turn have work to do to develop a deeper understanding of the simple conceptual ideas we’ve introduced here.

Let’s first state our working definition of “limit” that we can apply to any function:

The limit of the function $f(x)$ as x approaches a, written

\[\lim_{x \to a} f(x) = L\]

is a number L (if one exists) such that $f(x)$ is as close to L as we want
whenever x is sufficiently close to a.

For example, for the function we considered at the top of this screen, $f(x) = \dfrac{x^2\, -\, 4}{x\,-\,2},$ we have
\[\lim_{x \to 2} f(x) = 4,\] which, once again, means, “We can get as close to $f(x)$ = 4 as we’d like by being sufficiently close to x = 2.”

With our working definition of limits in mind, let’s consider a few questions to further develop your understanding. The first question helps drive home a key distinction:

Introductory Limits Question #1: Function's value versus limit value

Consider the function
\[g(x) =
\dfrac{x^2\, -\, 4}{x\,-\,2} & \text{for } x \ne 2 \\
\quad 5 & \text{for } x = 2
\end{cases}\] An altered version of the function we're using for our limits introduction: graph of the line y = x+2 with a hole at (2,4)), with the addition of the single point (2, 5).

The limit of a function at a point and the function’s value at that point need not be the same value.
As Question 1 illustrates, the limit of a function at a point and the function’s value at that point need not be the same value.

Let’s consider a similar question.

Limits Introduction Question #2: Another function's value versus limit value

Consider the function $h(x) = 6\, -\, (x\,-\,5)^2.$
Limits introduction question #2: Graph of the curve y = 6 - (x-5)^2. The function's value at x=7 is h(7)=2.

As we see in Question 2, the limit of a function at a point can simply equal the function’s value at that point, and often does. None the less, it is critical that you recognize that the two quantities $h(3)$ and $\displaystyle{ \lim_{x \to 3}h(x)}$ are distinct. They may be equal, or, as we saw in Question 1, they may not be, depending on the particular function and the particular input-value of interest.

So far on this screen we’ve used graphs to see easily what a function’s limit is at a given point. You of course won’t always have such a graph available, and there are other ways to provide data about a function. For instance, on exams you should expect questions that present you a table of values intead. The next question illustrates.

Limits Introduction Question #3: Limit from a table

The table below shows some input and output values for a particular function $k(x).$



Smooth curve with a hole at x=3. By eye, the hole has y-value 0.25. Regarding Question #3, you might be thinking that you cannot know for sure that the answer is $\displaystyle{\lim_{x \to 3}k(x) = 0.25 }.$ If so, you are correct! We do not have enough information from the table alone to draw a definite conclusion. But when a question says something like, “Based on the values in the table, …”, it means for you to assume that the function is smooth and doesn’t jump around wildly, and instead just happens to have a hole at some input value (here, $x=3$). So in your head craft a mental picture of a curve that has decreasing output values as you pass through input values a little below $x=3$ to a little above $x=3$, and then answer based on that mental picture, essentially filling in the output-value of that hole since that’s the value we could get as close to as we wanted by being sufficiently close to $x=3.$ In case it helps, the figure to the right is a graph of the function from which the table was crafted.

Common misunderstandings of “limits”

Before we leave this screen, let’s address a few common ideas students have when they’re first learning about limits in Calculus.

Limits Introduction Question #4: The limit depends on function's value at the point of interest?
Limits Introduction Question #5: A function can exceed the limit's value?

We wanted to raise the true/false question right here on this introductory screen because the word “limit” itself has everyday, non-math usage that we should acknowledge, things like “speed limit,” “I have limited free time,” “I’m at my limit.” That is, in everyday usage a “limit” is something that you typically do not (or should not, anyway) exceed. The mathematical meaning is different, as we can see from our current working definition:

The limit of the function $f(x)$ as x approaches a
is a number L (if one exists) such that $f(x)$ is as close to L as we want
whenever x is sufficiently close to a.

From this definition, we can certainly have values above, and below, the limit-value at a particular point. Hence just keep in mind that you should be careful when applying your everyday ideas of “limits” to our new mathematical usage.

Figures illustrating common uses of the word limit that do not apply to our mathematical definition: Big red-X through picture of height-limit sign, speed-limit sign, limited-supply sign, and mother saying to child I am at my limit!

Note that while our working definition of “limit” (repeated immediately below) is conceptually appealing and, for many, easy to understand, the phrases “as close to” and “sufficiently close” are vague and hence open to different interpretations by different people. That is, while a fine place to end this introduction to limits, those phrases are not mathematically precise. We’ll start to make them more precise on the next screen.

Do you have questions or comments about anything on this screen? Please head over to the Forum and post them there!

The Upshot

  1. In our introduction to limits, we have developed a working definition:

    The limit of the function $f(x)$ as x approaches a, written
    \[\lim_{x \to a} f(x) = L\]is a number L (if one exists) such that $f(x)$ is as close to L as we want

    whenever x is sufficiently close to a.

  2. Common, everyday ideas of “limit” do not apply to our mathematical notion of “limit.”

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