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## Calculus 1

### Unit 1: Lesson 3

Estimating limits from tables

# One-sided limits from tables

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.C (LO)
,
LIM‑1.C.5 (EK)
When we read a one-sided limit from a table, we are only interested in the value the function approaches from a single direction (either left or right).

## Want to join the conversation?

• When can you tell if the limit of a one-sided function does not exist? I am having trouble figuring that out.
• A limit is defined as the value of a function f(x) as x approaches some c value from both sides of said c value. A one-sided limit is the same as a regular limit, but it only requires one side of the function to be approaching the c value. One-sided limits may not exist in the following cases:
-The function goes to infinity (a vertical tangent) at the c value from that side
-The function doesn't approach the c value from that side

The one sided limit still exists in the following cases:
-The limit from the other side is different
-The actual value of the function at the point c doesn't exist (hole), but the one side of the function still approaches a value
-The actual value of the function jumps at the point c, but the one side of the function still approaches a value
-A discontinuity in the function that doesn't affect the ability to determine the limit

Hope this helps!
• Will we learn how to find out the actual limit of a function later in calculus, or is that not possible?
• You will learn how to find the actual limit. It's later on in this unit.
• Wouldn't it be more reasonable to assume that lim(x->-2^-) f(x)=-2500?
Because -20*5=-100, -100*5=-500, -500*5=-2500
• The thing is, why did you stop there? Perhaps -2500 is the value at -2.0001. Maybe there's another value at -2.00001, say -2500 * 5 = -12500. Is that the limit? Or will it multiply by five again if we try a closer number? Just how close do we need to get before we're certain?

If the values started to slow down at some point, then it might be a reasonable estimate. For example:
-2.0001 = 2499
-2.00001 = 2499.5
-2.000001 = 2499.75
In this case, yes, -2500 is probably a good guess. But working just from the table we're given, the numbers appear to simply be increasing exponentially as we approach -2, and for that reason we can assume it probably is unbounded.
• How to know that the function is unbounded ?
• consider x approaching -2 and y=f(x). We see that as the function grows closer to -2 (-2.1,-2.01,-2.001, -2.00000001, etc...) f(x) falls exponentially (-100, -500, etc..). Plot this on a graph and you'll notice the function is unbounded at x= -2 and that the limit approaching from the left is infinity (or doesn't exist).
• What I must do to be 100% sure about the limit of a function?What is the method for finding the limit of a function with absolute certainty?
• This is a good question. Using tables only suggests what the limit might be, but does not give the limit with certainty. However, limits (if they exist) can be found with certainty by using algebraic techniques such as direct substitution, factoring & reducing, multiplying top and bottom by a conjugate, using trigonometric identities, and using L’Hopital’s rule.
• Wait, I'm confused, why was the limit to the left negative 4? Shouldn't it be it doesn't exist, since -4 was an f(x) value?
(1 vote)
• I don't think the actual value of the function at the specified point is needed in order to determine the limit. You only need values that get closer and closer to that point. Therefore, it is redundant that -4 is a value of f(x) as it does not affect the limit.
• If a one-sided limit is unbounded, do you say it is "unbounded" as the answer to the limit or "does not exist"
(1 vote)
• Usually it's enough to say that the limit doesn't exist, but if we need to specify why the limit doesn't exist, then we would have to say that it's because the function grows unboundedly.

This applies to one-sided and two-sided limits alike.
• On the second example on this timestamp: 5.00 confused me. Is it unbounded because because f(-2.002) (which is -500) is larger than f(2) (which is -4)? Or I am missing something?
(1 vote)

The idea is that, as you get closer and closer to -2 without actually getting there, it seems like the function is becoming exponentially larger. Notice how in the table, we can get really close, and the trend seems to stick.

Essentially, don't connect the dots on the table, try to sniff out patterns.

Hope this helps.
(1 vote)
• i dont understand how to solve these. can someone explain
(1 vote)
• It basically gives you a table with values on either side of the "targeted number" in the limit. For example, if it was lim x-> 3 f(x) then you would be given values smaller and bigger than 3 that are very close to it, like 2.999999 and 3.000001 and given these values you need to determine if both the one sided limits approach the same value. Lets say at x = 2.9 f(x) = 5.9 and at x=2.999 x = 5.999 then the one sided limit from the LEFT side is approaching 6 (it gets closer and closer to 6 as you get closer and closer to the x-value "3" from the LEFT), and say at x=3.1 f(x) = 6.1 and at x=3.001 x = 6.0001 then you can say that the one sided limit from the right is also approaching 6, and since both the one sided limits approach the same value, the limit of f(x) as x approaches 3 = 6. If both the one sided limits approached different values, we would say that the limit does not exist. I hope this helped, let me know if something i said was unclear or confusing
(1 vote)
• Hello! Great video limits started to getting so clear for me ! The only thing i dont understand is when you can say that the limit of a function is negative or positive infinity and when you cannot?
(1 vote)
• I'd recommend you explore a graph of the function y = 1/x as an example of a function whose limit approaches ∞ from the right and -∞ from the left, as the function approaches zero.

In general, any rational function whose denominator tends to zero is one type of function whose limit will tend to infinity. There are others.
(1 vote)

## Video transcript

- [Instructor] The function F is defined over the real numbers. This table gives select values of F. We have our table here. For these X values, it gives the corresponding F of X. What is a reasonable estimate for the limit of F of X as X approaches one from the left? So pause this video and see if you can figure it out on your own. Alright, now let's work through this together. So the first thing that is really important to realize is when you see this X approaches one and you see this little negative superscript here, this does not mean approaching negative one, so this does not mean negative one. Sometimes your brain just sees a one and that little negative sign there, and you're like oh, this must be a weird way of writing negative one, or you don't even think about it, but it's not saying that. It's saying, this is saying, let me put a little arrow here, this is the limit of F of X as X approaches one from the left, from the left. So from the left, how do we know that? Well that's what that little negative tells us. It tells us we're approaching one from values less than one. If we were approaching one from the right, from values greater than one, that would be a positive sign right over there. So let's think about it. We want the limit as X approaches one from the left, and lucky for us on this table, we have some values of X approaching one from the left. 0.9, which is already pretty close to one, then we get even closer to one from the left. Notice, these are all less than one, but they're getting closer and closer to one. And so what we really wanna look at is what does F of X approach as X is getting closer and closer to one, from the left, from the left. And a key realization here is, if we're thinking about general limits, not just from one direction, then we might wanna look at from the left and from the right, but they're asking us only from the left, so we should only be looking at these values right over here. In fact, we shouldn't even let the value of F of X at X equal one confuse us. Sometimes and oftentimes, the limit is approaching a different value than the value of the function at that point. So let's look at this. At 0.9, F of X is 2.5. When we get even closer to one from the left, we go to 2.1. When we get even closer to one from the left, we're getting even closer to two. So a reasonable estimate for the limit as X approaches one from the left of F of X, it looks like F of X right over here is approaching two. We don't know for sure, that's why they're saying, what is a reasonable estimate. It might be approaching 2.01 or it might be approaching 1.999. On Khan Academy these will often be multiple choice questions, so you have to pick the most reasonable one, it would not be fair if they gave a 1.999 as a choice and 2.01, but if you were saying, hey, maybe this was approaching a whole number, then two could be a reasonable estimate right over here. Although it doesn't have to be two, it could be 2.01258, it might be what it is actually approaching. So let's try another example here. Here it does look like there's a reasonable estimate for the limit as we approach this value from the left. So now, it says the function F is defined over the real numbers. This table gives select values of F, similar to the last question. What is a reasonable estimate for the limit as X approaches negative two from the left? So this is confusing. You see these two negative signs. This first negative sign tells us we're approaching negative two. We wanna say, what happens we're approaching negative two, and we're gonna approach once again from the left. So lucky for us, they have values of X that are approaching negative two from the left, so this is X approaches negative two from the left, so that is happening right over here. So that's these values. So notice, this is negative 2.05, then we get even closer, negative 2.01, then we get even closer, negative 2.002, and these are from the left because these are values less than negative two, but they're getting closer and closer to negative two. And so let's see, when we're a little bit further, F of X is negative 20, we get a little bit closer, it's negative 100, we get even a little bit closer, it goes to negative 500. So it would be reasonable, and we don't know for sure, this is just giving us a few sample points for this function, but if we follow this trend, as we get closer and closer to negative two, without getting there, it looks like this is getting unbounded. It looks like it's becoming infinitely negative. And so technically, it looks like this is, I would write this is unbounded, and so if this was a multiple choice question, technically you would say the limit as X approaches negative two from the left does not exist, does not exist. If someone asked the other question, if they said, what is the limit as X approaches negative two from the right of F of X, well then you would say, alright, well here are values approaching negative two from the right, so this is X approaching negative two from the right, right over here. And remember, when you're looking at a limit, sometimes it might be distracting to look at the actual value of the function at that point, so we wanna think about what is the value of the function approaching, as your X is approaching that value, as X is approaching, in this case, negative two, from the right. So as we're getting closer and closer to negative two from values larger than negative two, it looks like F of X is getting closer and closer to negative four, which is F of negative two, but that actually seems like a reasonable estimate. Once again, we don't know absolutely for sure, just by sampling some points, but this would be a reasonable estimate. And in general, if you are approaching different values from the left than from the right, then you would say, a that point, the limit of your function does not exist, and we have seen that in other videos.