Geometric View - One Sided Limits

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Visual introduction to One-sided Limits

One-sided limits

Let’s say we are trying to determine the limit of $f(x)$ as $x$ approaches $5$ . It turns out considering $x$ values “near $5$ ” is a little too vague. Instead, it helps to break the neighborhood around $5$ into two regions, the values just to the left of $5$ (i.e. slightly less than $5$ or slightly more negative than $5$ ) and the values just to the right of $5$ (i.e. slightly larger than $5$ or slightly more positive than $5$ ). This is the concept of a one-sided limit!

Geometric definition of one-sided limits A left limit, also called a limit “from the left” is the value being approached by $x$ -values approaching our point of interest (in the case above, $5$ ) from below (a.k.a. values to the left on the numberline).

A right limit, also called a limit “from the right” is the value being approached by $x$ -values approaching our point of interest (in the case above, $5$ ) from above (a.k.a. values to the right on the numberline).

We will introduce the formal notation in the next section, but for now let’s see some examples.

Recall the graph from our previous section:

1 : Here, if we consider the values as $x$ approaches $0$ from the left, we can see that $f(x)$ approaches...

$0$ $1$ Cannot be determined, since $f(0)$ is not defined. Any number is valid since the function is not defined at $x=0$ and thus we can make it anything.

From this setting we come to our definition of a limit.

Intuitively, the limit of $f(x)$ as $x$ approaches $a$ is $L$ , if the value of $f(x)$ is as close as one wishes to $L$ for all $x$ sufficiently close, but not equal to, $a$ .

2 : Consider the following graph of $y=f(x)$

Use the graph to evaluate the following. Write DNE if the value does not exist.

(a): $f(-2) =\answer {1}$
(b): The limit as $x$ approaches $-2$ is $\answer {1}$
(c): $f(-1) =\answer {2}$
(d): The limit as $x$ approaches $-1$ is $\answer {2}$
(e): $f(0) =\answer {-2}$
(f): The limit as $x$ approaches $0$ is $\answer {0}$
(g): $f(1) =\answer {DNE}$
(h): The limit as $x$ approaches $1$ is $\answer {-2}$

Limits might not exist

Limits might not exist. Let’s see how this happens.

Consider the graph of the floor function, $f(x) = \lfloor x\rfloor$ .

Explain why the limit as $x$ approaches $2$ of $f(x)$ does not exist.

The function $\lfloor x \rfloor$ , called the “floor” function, is the function that returns the greatest integer less than or equal to $x$ . Now recall that, intuitively, we know a limit exists at $x=2$ if there exists some number $L$ so that $\lfloor x\rfloor$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $2$ . So let’s examine $x$ near, but not at, $2$ , and see if we can find such an $L$ ?

If this limit exists, then we should be able to look sufficiently close, but not at, $x=2$ , and see that $f$ is approaching some number. Let’s look closer at the portion of our graph near our point:

If we allow x values on the left of 2 to get closer and closer to 2, we see that $f(x) = 1$ . However, if we allow the values of $x$ on the right of 2 to get closer and closer to 2...

We see that $f(x) = 2$ for all these values just to the right. We cannot find a single number $L$ that $f(x)$ approaches as $x$ approaches 2! In essence, since the left limit ( $1$ ) and the right limit ( $2$ ) are not going to the same place, the limit at $2$ cannot exist, because we can’t get a single value $L$ for our definition.

You may have noticed that in our example above, the function is also discontinuous at that point. It turns out this is true in general! We will explore the relationship between continuity and limits a little later (including finally giving a formal definition of continuity, rather than our intuitive definition we’ve had since before precalculus!) but for now we state the following useful result:

Continuity implies limit exists If a function $f(x)$ is continuous at the $x$ value $a$ , then the limit at $a$ exists, and equals $f(a)$ .

If you don’t recall what it means to be “continuous at $a$ ”, click the blue arrow to the right!

You can intuitively think of a function being continuous at a point, as meaning that there isn’t a discontinuity at that point. You can review types of discontinuities and what they mean here (You should open the link in a new tab!).

The actual justification/proof for the above theorem will be covered when we dive deeper into continuity, but this result will allow us to investigate many useful examples in the meantime.