If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 1

Lesson 4: One-sided limits

# 1-sided vs. 2-sided limits (graphical)

Sal explains the relationship between the 1-sided limits and the 2-sided limit of a function at a point, using a graphical example where the 1-sided limits exist but the 2-sided limit doesn't. Created by Sal Khan.

## Want to join the conversation?

• How is it possible, that a limit can be both a value that's filled in, and a value that's not? I mean the "including" symbol on the graph. If we include 3 too, then is it really a limit? And if we didn't, how can it be the same limit? •   Good question. When we talk about a limit, we're asking what happens to a function value as we move arbitrarily close to a particular point. We don't care what the value is at that point. So in this example, the limit as x approaches 3 from the left is 4 even though the value at x=3 is 1. The limit is 4 because when we approach from that direction it gets closer and closer to 4 and we can make it just as close to 4 as we want, even though we can't make it exactly equal to 4. That's the essential meaning of a limit: we can get as close as we want, even if we can't get all the way there. Working from the right we get a different limit in this case, and it's still considered a limit even though the function is defined at that point and the definition matches the limit. In other words, even though we know f(3)=1, we're still interested in knowing the answer to the distinct question, what number (if any) does the function approach as x approaches 3.
• Is there any mathematical symbol for "does not exist" that can be used in these kinds of problems?
I tend to prefer symbols, since in my opinion they are often more convenient and reader-friendly than words. • Also, how is it possible to have two values that are filled in on the same x value? (Inspired by Aggelos Giannoudakis) • In all these videos, I had a doubt. What does colored and empty dot stand for? What is the difference? • What happens when you have to determine f(0) or f(2), f(3), etc? • Hi Natalie,

A function is a set of rules that gives a single output for a unique input. This is usually shown in the form f(x) = some expression involving x. This explains what happens to the input in order to give you an output.

For example, f(x)=3x describes a function that gives 3 times the input as the output. To specify the input, you replace every appearance of x with your input. This gives you the specific output that corresponds to a specific input to the function.

Using your examples with the function f(x)=3x:

f(0) = 3*0 = 0
f(2) = 3*2 = 6
f(3) = 3*3 = 9
(1 vote)
• Would the concept of a limit finally show the solution to Zeno's paradox, which states that you can get infinitely close to passing a turtle with a head start, but it will always surpass your distance? Or that, given a finite series of half-distance jumps, you would reach the other side of the room? • So,(I just want to make sure) a limit still won't exist even if the function is defined at that x value?
(like the function in the video,and the ones in the previous videos) • So what's the difference between a two-sided limit from a graph and a one-sided limit from a graph? From what I can tell, the process is exactly the same as the process explained in the "One-sided limits from graphs" video. • That's a great question - and later videos will answer it.
Essentially, for us to say the limit, L, of some function, f, actually exists as x approaches some value, say c, then we must have it that both the one sided limits must be present and equal the same value, L, otherwise we say the the limit of f as x approaches c does not exist.

This is super important for what is to follow - differential calculus - the next section in Khan, where you will delve deeper into the concept of a limit before you learn differentiation.

Remember finding the slope of a line back in algebra? That required that you know two points on the line to calculate it. Differentiation allows you to find the slope of a curve (and lines) at a single point on the curve, except that now you will call the slope the derivative. It is way fun!  