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# One-sided limits from graphs

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.C (LO)
,
LIM‑1.C.1 (EK)
,
LIM‑1.C.2 (EK)
,
LIM‑1.C.3 (EK)
,
LIM‑1.C.4 (EK)
A one-sided limit is the value the function approaches as the x-values approach the limit from *one side only*. For example, f(x)=|x|/x returns -1 for negative numbers, 1 for positive numbers, and isn't defined for 0. The one-sided *right* limit of f at x=0 is 1, and the one-sided *left* limit at x=0 is -1. Created by Sal Khan.

## Want to join the conversation?

• why is a limit not defined if the left-handed limits and the right-handed limts are unequal?? •  You could think about it like this: as you approach some point on the graph from opposite directions and you are approaching different y values, what would you say that the limit is? Is it one value? Is it the other? Well, it's both, depending on which direction you're coming from. So we say that there is no single 'ordinary' limit.
• I understand how quadratic equations, conic sections, and sine waves look on a graph and they're all very smooth. Is there ever a single equation that, when graphed, would look so broken and crooked as this first graph? Any of these graphs, for that matter. •   Do you mean graphs with discontinuities?

Many graphs that aren't piece-wise functions have discontinuities.
y=1/x
Here, there is a discontinuity at x=0. From one side, the graph goes down to
-Infinity while from the other side, the graph goes up to Infinity. This is called an infinite discontinuity. Can you ascertain why?

y=x/x
This is like the piece-wise function y=1 for x does not equal 0 (x/x=1) and y is undefined for x=0 (0/0 is indeterminate) (put your mouse art x=0 and see what y is). This is a removable discontinuity because all we did was remove one point from the graph and let the graph be normal everywhere else.

Let's say that ⌊x⌋ is the greatest integer less than or equal to x. Yes, this is a mathematical function.
y=x-⌊x⌋
I couldn't find a graph of this...
Let's try to visualize this. At every integer x, ⌊x⌋ is x. At every integer x,
x-⌊x⌋=x-x=0. However, from the left side, the function will be approaching 1. This is because the distance between two consecutive integers is 1. From the left side of integers, the numbers will be close to an integer, but the ⌊x⌋ of x will be all the way at the last integer. This means that x-⌊x⌋ is approaching x-(x-1), or 1. However, when it's supposed to get to 1, it jumps down to 0 every time. These discontinuities are called jump discontinuities because you jump from one place to another.

I hope this helps you understand discontinuities outside of piece-wise functions!
• This is hard. Im a year 8. Should I be learning this yet? I find it v. complicated and I dont understand it. The video helped a little. Thanks :) •  Calculus is usually presented in year 11 or year 12, though different countries have their own standards.

Before attempting calculus, you need to have mastered algebra, geometry and trigonometry. In calculus, it is assumed that you know those topics very well. If you haven't studied those topics, this material is going to be too advanced for you to understand.
• • How come the limit of f(x) is 5 in . Shouldn't it be -5 because both limits from both directions are -5? • You are correct! Good catch.

Also, notice as you watch the video at that there is a small box in the lower right-hand corner of the video that says:

> Correction: The general limit is -5.

Have a great day! PS Minecraft is not so bad...
• Why is limit the same whether the graph has an open circle [as in the endpoint is not included in the solution] and when it has a closed one? Thanks • Are you familiar with Zeno's (ancient Greek philosopher) paradox? I find it useful to explain this concept of limits. Zeno argued that motion was impossible since, for example if you shot an arrow at a target the arrow would first have to travel half the distance to the target, then half the remaining distance, the half again and again and again, ad infinitum and thus never reach the target since the arrow always has to traverse the remaining half distance. Now in our physical world, that doesn't happen (and why is another discussion) BUT in the abstract math world, THIS IS TRUE. If you tell me you are x distance away from something, I can half the distance and be even closer than you, NO MATTER WHAT DISTANCE x you choose. With limits we are saying that no matter how close you want to get to the limit value, you can ALWAYS get closer - it doesn't matter if the limit value endpoint is included or not, you will never 'get' to it anyway since you can always half the distance your are from it. But since in either case (endpoint value included or not) you can keep getting as close as you want to it, the limit is the same. Hope that helps :)
• What is actually meant by limit exist , i mean literally what is the significance when we say limit exists?
(1 vote) • Do you mean a rigorous mathematical definition or an explanation in simpler terms?

For the simple explanation:
A limit is said to exist for some function f(x) for some value c if f(x) clearly gets closer and closer to some finite value as x gets closer and closer to c. This means that you get the SAME value whether x is less than c and increasing toward c OR x is greater than c and and decreasing toward c. If all these conditions are met, the limit is said to exist. If any is not met, the limit is said to fail to exist or just "does not exist".

NOTE (and this is EXTREMELY important to understand): what the function is approaching as f(x) approaches x=c does NOT have to be the same thing as f(c). If there is a sudden change in f(x) at x=c, then the limit (if it exists) would be what f(x) "should have been" had the sudden change not been there. Also, a limit can SOMETIMES exist where the function itself is undefined.

As for the formal definition of a limit, I will defer to one of the professional mathematicians / math majors because that gets very tedious and I don't want to risk making a mistake.
• If we have
f(x)=x,x<5,
then does the limit of x approaches 5 exists?   