Calculus, all content (2017 edition)
- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- One-sided limits from graphs
- One-sided limits from tables
- 1-sided vs. 2-sided limits (graphical)
- Limits of piecewise functions: absolute value
- Connecting limits and graphical behavior (more examples)
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.
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- 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?(74 votes)
- 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.(117 votes)
- 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.(12 votes)
- This symbol means "Does not exists":
- Also, how is it possible to have two values that are filled in on the same x value? (Inspired by Aggelos Giannoudakis)(4 votes)
- That's impossible for functions, as the definition of a function means that for every input there is only one single output. Or, for every x value, there is only 1 f(x) or y value. BUT, for every y value, there could be more than 1 x value.(6 votes)
- In all these videos, I had a doubt. What does colored and empty dot stand for? What is the difference?(2 votes)
- The colored dot means that the value is included in the function/line. The empty dot means that that exact value is NOT included in the function/line.(8 votes)
- What happens when you have to determine f(0) or f(2), f(3), etc?(5 votes)
- 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?(4 votes)
- I think they are really more of an example of an infinite series, but of course you can think of an infinite series as the limit of a finite series as the number of terms goes to infinity.(3 votes)
- 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)(2 votes)
- 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.(3 votes)
- 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!(2 votes)
- Do we always use integers for limits?(2 votes)
- No. The limit problems that are given use integers. However, fractions, decimals, etc. can be used as well.(4 votes)
- Would it be correct to say that a limit can and cannot exist at the same time, assuming that the limits can be 1-sided and 2-sided?(1 vote)
- You have to talk about limits in terms of what they're approaching and from what direction. It doesn't make sense to say limits do and do not exist at the same time. However you can have one-sided limits that exist and a double-sided limit that does not exist. The double-sided limit only exist if both one-sided limits are the same. For example look at the unit step function. As x approaches 0 from the left, the limit approaches 0 (the left-sided limit exists) and when x approaches 0 from the right, the limit approaches 1 (the right-sided limit exists). Because the two one-sided limits are approaching two different values, namely 0 and 1, the double-sided limit does not exist.(5 votes)
So we have the graph of a function right over here, and we want to think about what does the limit of f of x as x approaches 3 appear to be? And to do that, let's just think about the limit as x approaches 3 from values less than 3, and from values greater than 3. So let's first think about the limit of f of x as x approaches 3 from values less than 3. So this little negative superscript says we're going to approach 3 from below 3. From 1, 2, 2.5, 2.99, 2.999. So if we approach-- so this is 3 right over here. And we're going take the left handed, or the left sided limit. We're going to approach 3 from this direction first. So when x is 0, f of x is there. When x is 1, f of x is there. When x is 2, f of x is there. When x is 2 and 1/2, f of x is at 5. When x is at-- looks like roughly 2 and 3/4, we get to 4. Looks like about f of x gets to 4.5. And so it looks like as x approaches 3 from values less than 3, it looks like our function is approaching 4. So I would say it looks like the left sided limit of f of x as x approaches 3 is 4. Now let's do the same thing for the right hand side. So the limit of f of x as x approaches 3 from values larger than 3. So notice when x is equal to 5, our f of x is up here. When x is equal to 4, f of x is here. When x is 3 and 1/2, it looks like we're a little under 2 for f of x. And it looks like we're getting closer and closer as x approaches 3 from the positive direction, or from the right side, it looks like f of x is getting closer and closer to 1. So I would estimate, based on this graph, that the limit of f of x as x approaches 3 from the positive direction is equal to 1. Now we have an issue. In order for this limit to exist, we have to get the same value as we approach from the left hand side and the right hand side, but it's clear that we are not approaching the same value when we go from the left hand side as we do when we go from the right hand side. So this limit right over here does not exist. Does not exist. The only way that this would have existed is if we got the same value for both of these, and then the limit would be that value. But we're clearly not getting the same value.