Limits: Notation, One-sided Limits, and Formal Definitions

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We introduce the notation and formal algebraic definitions for limits.

Video Lecture

Limit Notation

You may have noticed at this point that it can get cumbersome and tiring to keep saying the phrase “the limit of $f(x)$ as $x$ approaches $a$ ” and mathematicians are inherently lazy, so we have defined a notation that encapsulates this entire phrase, which we present next.

Limit Notation If the limit of $f(x)$ as $x$ approaches $a$ is $L$ (according to our intuitive definition), then we write this as; $\lim \limits _{x\rightarrow a} f(x) = L$

This notation breaks down the limit information we need into neat little blocks, which we will pull apart and parse here.

The $\lim$ at the start tells us we’re dealing with a limit (I said mathematicians were lazy, not creative!).
Underneath (or sometimes a subscript in front) of the $\lim$ is “ $x \rightarrow a$ ”. This block is telling us that the $x$ value is approaching the value $a$ , which is where the limit is happening. In other words $a$ is the “ $x$ -value of interest”.
Notice that the whole block $\lim \limits _{x\rightarrow a}$ is saying “the limit, as $x$ approaches $a$ of...” Thus the $f(x)$ is the thing that the limit is being computed for. You can think of the $\lim \limits _{x\rightarrow a}$ as full brick of notation that is then applied to a function. This means it makes no sense to write this thing on its own, it inherently must be applied to something.
Finally, the $= L$ part is just telling us that the thing on the left of the “ $=$ ” sign (the limit as $x$ approaches $a$ of $f(x)$ ) is the same value as $L$ , which is the very wordy way of saying that the limit is $L$ .

The notation above is pretty dense and something that we will be using a lot. Take a few moments to really digest what the notation means. What would it mean to change the $a$ value? How about the $L$ value, or $f(x)$ ? Make sure you understand the notation well before moving forward as we will be adding a couple more notational bits which are subtle, but very important.

One-sided Limits

Thus far we have used tables and graphs to determine limits, and we’ve discussed the possible up and downsides of both. Our next goal is to formalize the analytical idea of the limit; which requires the introduction of the one-sided limit. Recall that it can be helpful to break the region near our point of interest into the (nearby) region to the left of $a$ and the (nearby) region to the right of $a$ . We formalize the notation for these left and right limits next.

For the function $f$ , $L$ is the limit from the right as $x$ approaches $a$ , written $\lim _{x\to a^+} f(x) = L,$ if the value of $f(x)$ is as close as one wishes to $L$ for all $x>a$ sufficiently close to $a$ .

Similarly, for the function $f$ , $L$ is the limit from the left as $x$ approaches $a$ , written $\lim _{x\to a^-} f(x) = L,$ if the value of $f(x)$ is as close as one wishes to $L$ for all $x<a$ sufficiently close to $a$ .

Recall that a limit exists at $a$ and equals $L$ if; for $x$ values sufficiently close to $a$ , $f(x)$ gets as close as we want to $L$ . But notice this is equivalent to saying that both the left and right limits are equal to the same value; namely to $L$ . In other words, a limit exists if, and only if, both one-sided limits exist, and have the same value! We formalize this idea in the following theorem.

A limit $\lim _{x \to a} f(x)$ exists if and only if

$\lim _{x \to a^-} f(x)$ exists
$\lim _{x \to a^+} f(x)$ exists
$\lim _{x \to a^-} f(x) = \lim _{x \to a^+} f(x)$

In this case, $\lim _{x \to a} f(x)$ is equal to the common value of the two one sided limits.

If the theorem above doesn’t look like what you remember from another class or textbook, click the blue arrow to the right for a note on how this is often presented in other textbooks/lectures!

Often in literature (like textbooks or notes) you will see the above theorem written as the following:

The limit $\lim \limits _{x\rightarrow a} f(x)$ exists and equals $L$ if $\lim \limits _{x\rightarrow a^+} f(x) = L = \lim \limits _{x\rightarrow a^-} f(x)$

Notice that this is actually the same thing, but this definition is a little less clear. Specifically

$\lim \limits _{x\rightarrow a^+} f(x)$ and

$\lim \limits _{x\rightarrow a^-} f(x)$ can only equal L (and each other) if they exist. So this definition is hiding the fact that you need to check if the one sided limits exist in the first place by burying it as an assumption into the equality. You can use either definition (they are equivalent after all), but remember that this second definition has more going on than first meets the eye!

1 : Suppose you are taking a quiz or an exam and one of the problems gives you a function $f(x)$ , then asks you to figure out if $\lim \limits _{x\rightarrow a} f(x)$ exists, and if it exists, to calculate its value.

You should start by...

Graphing the function. Determining the value of the function at $a$ . Calculate both one-sided limits of $f(x)$ at $x=a$ . Calculate one of the one-sided limits of $f(x)$ at $x=a$ . calculate $f(x)$ at $0$ .

1.1 : Next you...

Conclude that the limit exists. Check to see if the two one-sided limits equal each other. Conclude the limit DNE. Calculate $f(a)$ . Conclude the value of the limit.

1.1.1 : If the limits agree (i.e. both limits equal some $L$ )...

You may conclude the limit DNE. You can’t conclude if the limit exists or not, nor its value. You can conclude the limit exists, but not its value. You can conclude the limit exists and its value.

1.1.1.1 : In particular the limit is equal to...

$0$ The value of the one-sided limit, i.e. $L$ . $f(a)$ . You can’t know from the information you have so far.

1.1.2 : If the limits don’t agree...

You may conclude the limit DNE (Does Not Exist). You can’t conclude if the limit exists or not, nor its value. You can conclude the limit exists, but not its value. yYou can conclude the limit exists and its value.

2 : Evaluate the expressions by referencing the graph below. Write DNE if the limit does not exist.

(a): $\lim _{x\to 4} f(x) \begin {prompt}=\answer {8}\end {prompt}$
(b): $\lim _{x\to -3} f(x)\begin {prompt}=\answer {6}\end {prompt}$
(c): $\lim _{x\to -3^-} f(x) \begin {prompt}=\answer {6}\end {prompt}$
(d): $\lim _{x\to 0} f(x) \begin {prompt}=\answer {DNE}\end {prompt}$
(e): $\lim _{x\to 0^-} f(x) \begin {prompt}=\answer {-2}\end {prompt}$
(f): $\lim _{x\to 0^+} f(x) \begin {prompt}=\answer {-1}\end {prompt}$
(g): $f(0) \begin {prompt}=\answer {-\frac {3}{2}}\end {prompt}$
(h): $f(-2) \begin {prompt}=\answer {8}\end {prompt}$
(i): $\lim _{x\to -2^+} f(x) \begin {prompt}=\answer {2}\end {prompt}$
(j): $\lim _{x\to -2^-} f(x) \begin {prompt}=\answer {6}\end {prompt}$
(k): $\lim _{x\to 2^-} f(x) \begin {prompt}=\answer {7}\end {prompt}$
(l): $\lim _{x\to 1} f(x) \begin {prompt}=\answer {3}\end {prompt}$

Optional Content

If you are interested in how mathematicians actually prove a limit’s value rigorously you can watch the following video on how mathematicians tackle the problem of testing all nearby points simultaneously. [URL to YouTube video, but not an embedded video.]