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1. Limits and Continuity

Introduction to Limits

1. Limits and Continuity

Introduction to Limits - Online Tutor, Practice Problems & Exam Prep

1

concept

Finding Limits Numerically and Graphically

Video duration:

6m

Video transcript

Welcome back, everyone. Here, we're going to be taking our very first look at limits. Now the first time that you come across a limit, they may seem a bit intimidating. But in finding a limit, all we're doing is looking at what a function is doing around some given value of x. Now we can do this easily by either looking at a graph or by creating a table of values.

Now here I'm going to show you both of these ways of finding limits and we'll also walk through some of the notation that you'll encounter when dealing with limits. So let's go ahead and jump in here. Let's first take a look at our limit notation. This is what you'll see anytime you're working with a limit and we can read this as the limit of f of x, our function, as x approaches c, the value for which we're taking the limit. Now looking at this limit here, this is asking us to find the limit of x squared, that's our function f of x, as x approaches 2.

That's our c value. Now our limit will always be equal to some value n, so let's actually find that value for this limit here. Now remember that I said in finding a limit, we want to look at what our function is doing around that given value of x. Now we can get a bit more specific here and say that a limit is actually the y value that a function f of x goes to as x gets really close to a given value. So here, we're looking for the y value that our function x squared goes to as x gets really close to 2.

Now what does that mean x gets really close to 2? Well, it means that we don't care what's happening when x is 2 but just when x gets really, really close to 2. So looking at our graph here, as x gets really, really close to 2 from either side, we want to look at what y value our function is going to. Now first looking at x from that left side closing in on 2, getting really, really close to it, I see that my function seems to be approaching a y value of 4. Now I can also think about this numerically and actually plug values in that are getting really, really close to 2 into my function.

So I can take numbers like 1.99 or closing in on 2 even further, 1.999. Now actually plugging these values into my function squaring them, here I get 3.96 and 3.996. Now here again, I can see that these values seem to be getting close to a y value of 4. Now let's look at that other side. Now closing in on 2, getting really, really close to it coming in from that other side, I see again that my function appears to be approaching or going to a value of 4.

Now, again, I can look at this numerically and actually plug values into my function that are closing in on 2 from that other side, like 2.01 and 2.001. Now again, plugging these values into my function, here I get 4.04 and 4.004. So again, from this side, I see that we seem to be approaching a y value of 4. Now again, here it does not matter what's happening when x is 2, but just as x gets really, really close to 2. Now here from both our graph and from our table of values, we see that the y value our function is going to as x gets really close to 2 is 4.

So our limit of x squared as x approaches 2 is 4. Now here we used 2 different ways of finding this limit. We've used a table and we used a graph. Now often you'll just be using one of these methods and problems will actually specify which method they want you to use. So let's work through a couple more examples together here.

Now this first problem asks us to find the limit of x+4 as x approaches 1 by creating a table of values. Now we want to look at values really, really close to 1 here. So let's first look at values closing in on 1 from this left side. So these values are at less than 1, but getting really close to it. So here, I want to look at values like 0.99 and 0.999.

Now plugging these values into my function x+4, here I get 4.99 and 4.999. So I can already see that this function seems to be going to a y value of 5. But let's also look at values really, really close to 1 closing in on it from the other side. So here values like 1.01 and 1.001, getting even closer there. Again, plugging these values into my function, here I get 5.01 and 5.001.

So here it is clear that we are going to a y value of 5. So the limit of x+4 as x approaches 1 is 5, and that's my answer here. Now something that you may have noticed for this problem as well as the one that we worked out above is that I could have just plugged my c value into my function. So here, if I would have just plugged 1 into my function x+4, I would have gotten 5, which is exactly what our limit is. So why am I using this language of x getting really, really close to 1 if I could have just plugged 1 in from the beginning and got my answer?

Well, the reason is that this is not always going to be true. You can't always just rely on plugging in that function value because your limit will not always be equal to the function value. So we can't just plug c in from the beginning. So with that in mind, let's look at this final example here. We want to find the limit of f of x as x approaches 3 using this graph here.

Now looking at our graph, we want to look at x getting really, really close to 3 from either side. Now see here looking at this graph coming in from this left side, I see as x gets close to 3, I'm approaching a y value of 1. Then coming in from the other side I see the same thing happening, as x gets really really close to 3 our function is going to a y value of 1. So the limit of f of x as x approaches 3 is 1. And here, if I would have just plugged 3 into my function or just looked at that function value on my graph, I would have gotten 4, which is a completely wrong answer.

That is not what my limit is because my limit is 1. So here it's very clear that our limit will indeed not always be equal to our function value. We need to look at x getting really, really close to c, not x actually being equal to c because we never know if our function value is going to be something entirely different like we saw on our graph here or if it even exists at that point. And it doesn't matter because we only care what's going on around it. So with this in mind, let's work through some more practice together.

Thanks for watching, and let me know if you have questions.

2

Problem

Problem

Find the limit by creating a table of values.
$\lim_{x\rarr0}-4x+2$

A

$-4$

B

$-2$

C

$0$

D

$2$

3

Problem

Problem

Find the limit by creating a table of values.
$\lim_{x\rarr2}3x^2+5x+1$

A

$1$

B

$10$

C

$23$

D

$21$

4

Problem

Problem

Find the limit by creating a table of values.
$\lim_{x\rarr1}\frac{x^2-4}{x-2}$

A

$0$

B

$3$

C

$-3$

D

$2$

5

Problem

Problem

Find the limit using the graph of $f\left(x\right)$ shown.
$\lim_{x\rarr1}f\left(x\right)$

A

$2$

B

$1$

C

$0$

D

Unable to determine

6

Problem

Problem

Find the limit using the graph of $f\left(x\right)$ shown.
$\lim_{x\rarr-2}f\left(x\right)$

A

$4$

B

$-2$

C

$-3$

D

Unable to determine

7

Problem

Problem

Find the limit using the graph of $f\left(x\right)$ shown.
$\lim_{x\rarr4}f\left(x\right)$

A

$2$

B

$-2$

C

$0$

D

Unable to determine

8

example

Finding Limits Numerically and Graphically Example 1

Video duration:

2m

Video transcript

Hey, everyone. Make sure that you go ahead and try this example completely on your own before checking back in with me. But here, we're asked to use the graph to estimate the value of each of our limits here. Now the first limit that we're asked to find is the limit of f(x) as x→−2. Now looking at our graph here, that tells us that we want to look at what our function is doing as x gets really, really close to negative 2 from either side.

So looking at our graph here, here is my x value of negative 2. And closing in on negative 2 from that left side, I see that my function seems to be approaching a value of 1. Also coming in from that right side, getting really, really close to negative 2, sort of closing in on it, I again see that my function seems to be approaching 1. So that tells me that my limit of f(x) as x→−2 is equal to 1. Now let's look at our next limit here.

Here, we're asked to find the limit of f(x) as x→0. So again, we want to look at values really, really close to 0 from either side, sort of closing in on it. Now, on our graph here, if we approach 0, getting really, really close to it from that one side, I see that my function seems to be approaching a value of 0. Then coming in from the other side again closing in on that x value of 0, I again see that my function seems to be approaching a value of 0. So that's what my limit is.

Now, here, you might get tripped up by that hole in the middle of our graph because when x is 0, it seems like our function is not defined. There's a hole in our graph. But remember that it does not matter what the actual value of our function is at 0, just what it's doing as x gets really, really close to 0. So don't let that hole freak you out and go ahead and just remember that your limit is as x approaches 0, not when it is 0. Now let's look at one final limit here.

We're asked to find the limit of f(x) as x→3. Now looking at 3 on our graph here, wanting to close in on it from either side, our x value of 3 I see here coming from this left side, then my function seems to be approaching a value of 1. Then coming in from that right side, I see the same thing happening. My function is approaching a value of 1. So my limit of f(x) as x→3 is equal to 1.

Now, looking at this graph, you might already start to see that we have a sort of crazy thing happening over here. And if I would have asked you to find the limit as x→4, you might get a little bit tripped up. Because from this side, I see that we're approaching 1. But from this side, we are not approaching 1. Now, I'm going to show you exactly how to deal with that in our next video. So I'll see you there.

Thanks for watching, and let me know if you have questions.

9

concept

One-Sided Limits

Video duration:

5m

Video transcript

When finding a limit, we look for the y value that our function is going to as x gets really close to a given value from either side. Now in looking at what our function is doing from either side of this x value, we've actually been finding what's called one-sided limits, which is something that you may be asked explicitly to find. Now because we've already been doing this, just not calling it one-sided limits, the only thing that's really new here is notation. So let's go ahead and jump right in. Now in working with a one-sided limit, we still write the limit of f of x, our function.

But instead of x just approaching c, we have x approaching c with this little negative sign. And we read this as the limit of f of x as x approaches c from the left. That's our left sided limit. Now for our right sided limit, we have the limit of f of x as x approaches c with this little positive sign from the right. Now in order to remember which sign represents which limit, you can think about your coordinate system because we know that our x values from the left are negative and our x values going to the right are positive.

So now that we've seen this notation, let's take a look at this function here and actually find some limits. Now let's start here with our left sided limit. So limit of f of x as x approaches 3 from the left. That's what that little negative sign tells me. Now first looking at my graph here, as we get really, really close to 3 coming in from that left side, I see that my function is approaching a y value of 1.

Now I can actually plug values that are getting really close to 3 from that left side into my function. And here as we're closing in on 3, I want to look at values like 2.99, 2.999. Now because this is a piecewise function and these values are less than 3, I know that my function here is equal to x minus 2. So plugging these values into my function, here I get 0.99 and 0.999. So just like I saw on my graph, I can see here based on these values that my function is going to a y value of 1.

So here, my left sided limit, limx→3−f(x)=1. Now let's take a look at our right sided limit here, the limit of f of x as x approaches 3 from the right. Now looking at our graph first here, as we get really, really close to 3 coming in from that right side, I see that my function is going to a y value of 4. And again, I can look at those values that are getting really, really close to 3 from that right side closing in on it, 3.01, 3.001, again, plugging those into my function, which here, since these values are greater than or equal to 3, my function here is just equal to 4. So again, based on these values, just like I saw on my graph, I can see that my function is going to a y value of 4.

So my right sided limit, limx→3+f(x)=4. So here we found our one-sided limits from the left and from the right. But what if I now asked you to find the limit of f of x as x approaches 3? Not from the left or from the right, but just as x approaches 3. Well, this might be a bit confusing because we just saw based on our one-sided limit that as x gets really, really close to 3 from either side, we see some different things happening.

And this tells us that because this function does not approach the same value from both sides, the limit actually does not exist or abbreviated DNE. Now this is something that you may have heard before. The limit does not exist. And here with this function, because as x approaches 3 from either side, we see some different things happening from one side going to a y value of 1 and from the other going to a y value of 4. This limit, does not exist.

Now here, we were able to find our one-sided limits. And because these one-sided limits were not equal to each other, that told us that our limit does not exist. So while you may be asked to explicitly find your one-sided limits, you may also be asked to use those one-sided limits to determine whether or not your regular limit, not your one-sided limit, exists or not. So let's go ahead and take a look at another example here. We want to use this graph to find the given limits.

Now the first limit that we're asked to find here is the limit of f of x as x approaches 1 from the left. That's what remember that that's what my little negative sign tells me here. So looking at my graph here, as x approaches 1 coming in from that left side, I see that my function is going to a y value of negative one. So here, my limit of f of x as x approaches 1 from the left, my left sided limit is negative one. Now let's move on to our next limit.

Here we have our right sided limit, the limit of f of x as x approaches 1 from the right. Now again, taking a look at our graph here, as x gets really, really close to 1 coming in from that right side, I see again that my function is going to a y value of negative 1. So here, my right sided limit is going to be the exact same as my left sided limit, negative one. Now let's look at one final limit here, the limit of f of x as x approaches 1. Just 1, not from the left or from the right.

So that means that we want to pay attention to what's happening as x approaches 1 from either side. But we already did that by finding our one-sided limits. Now here, since our one-sided limits are equal to each other, they're both negative one, that tells us that our limit here, the limit of f of x as x approaches 1, is also negative one, and we're done here. So when working with one-sided limits, if they are not the same, then your limit does not exist. But if they are the same, then your limit does exist.

So now that we've seen and put a notation to these one-sided limits and know how to use them to determine whether or not a limit exists. Let's continue practicing. Thanks for watching, and I'll see you in the next one.

10

Problem

Problem

Using the graph, find the specified limit or state that the limit does not exist (DNE).
$\lim_{x\rarr-2^{-}}f\left(x\right)$ , $\lim_{x\rarr-2^{+}}f\left(x\right)$ , $\lim_{x\rarr-2^{}}f\left(x\right)$

A

$\lim_{x \to - 2^{-}} - f (x) = 1 , \lim_{x \to - 2^{+}} f (x) = 1 , \lim_{x \to - 2} f (x) = 1$

B

$\lim_{x \to - 2^{-}} f (x) = 1, \lim_{x \to - 2^{+}} f (x) = - 1, \lim_{x \to - 2} f (x) = D N E$

C

$\lim_{x \to - 2^{-}} f (x) = 1, \lim_{x \to - 2^{+}} f (x) = 1, \lim_{x \to - 2} f (x) = D N E$

D

$\lim_{x \to - 2^{-}} f (x) = 0, \lim_{x \to - 2^{+}} f (x) = 0, \lim_{x \to - 2} f (x) = 0$

11

Problem

Problem

Using the graph, find the specified limit or state that the limit does not exist (DNE).
$\lim_{x\rightarrow0^{-}}f\left(x\right)$ , $\lim_{x\rightarrow0^{+}}f\left(x\right)$ , $\lim_{x\rightarrow0}f\left(x\right)$

A

$\lim_{x\rightarrow0^{-}}f\left(x\right)=0$ , $\lim_{x\rightarrow0^{+}}f\left(x\right)=0$ , $\lim_{x\rightarrow0}f\left(x\right)=0$

B

$\lim_{x\rightarrow0^{-}}f\left(x\right)=0$ , $\lim_{x\rightarrow0^{+}}f\left(x\right)=0$ , $\lim_{x\rightarrow0}f\left(x\right)=DNE$

C

$\lim_{x\rightarrow0^{-}}f\left(x\right)=-1$ , $\lim_{x\rightarrow0^{+}}f\left(x\right)=-1$ , $\lim_{x\rightarrow0}f\left(x\right)=DNE$

D

$\lim_{x\rightarrow0^{-}}f\left(x\right)=-1$ , $\lim_{x\rightarrow0^{+}}f\left(x\right)=-1$ , $\lim_{x\rightarrow0}f\left(x\right)=-1$

12

Problem

Problem

Using the graph, find the specified limit or state that the limit does not exist.
$\lim_{x\rightarrow4^{-}}f\left(x\right)$ , $\lim_{x\rightarrow4^{+}}f\left(x\right)$ , $\lim_{x\rightarrow4}f\left(x\right)$

A

$\lim_{x\rightarrow4^{-}}f\left(x\right)=1$ , $\lim_{x\rightarrow4^{+}}f\left(x\right)=1$ , $\lim_{x\rightarrow4}f\left(x\right)=1$

B

$\lim_{x\rightarrow4^{-}}f\left(x\right)=3$ , $\lim_{x\rightarrow4^{+}}f\left(x\right)=3$ , $\lim_{x\rightarrow4}f\left(x\right)=3$

C

$\lim_{x\rightarrow4^{-}}f\left(x\right)=3$ , $\lim_{x\rightarrow4^{+}}f\left(x\right)=1$ , $\lim_{x\rightarrow4}f\left(x\right)=DNE$

D

$\lim_{x\rightarrow4^{-}}f\left(x\right)=1$ , $\lim_{x\rightarrow4^{+}}f\left(x\right)=3$ , $\lim_{x\rightarrow4}f\left(x\right)=DNE$

13

Problem

Problem

Find the specified limit or state that the limit does not exist by creating a table of values.
$f\left(x\right)=\frac{1}{x}$
$\lim_{x\rightarrow1^{-}}f\left(x\right)$ , $\lim_{x\rightarrow1^{+}}f\left(x\right)$ , $\lim_{x\rightarrow1}f\left(x\right)$

A

$\lim_{x\rightarrow1^{-}}f\left(x\right)=0$ , $\lim_{x\rightarrow1^{+}}f\left(x\right)=0$ , $\lim_{x\rightarrow1}f\left(x\right)=1$

B

$\lim_{x\rightarrow1^{-}}f\left(x\right)=1$ , $\lim_{x\rightarrow1^{+}}f\left(x\right)=1$ , $\lim_{x\rightarrow1}f\left(x\right)=1$ $\lim_{x\rightarrow x}f\left(x\right)=1$

C

$\lim_{x\rightarrow1^{-}}f\left(x\right)=1$ , $\lim_{x\rightarrow1^{+}}f\left(x\right)=-1$ , $\lim_{x\rightarrow1}f\left(x\right)=DNE$

D

$\lim_{x\rightarrow1^{-}}f\left(x\right)=-1$ , $\lim_{x\rightarrow1^{+}}f\left(x\right)=-1$ , $\lim_{x\rightarrow1}f\left(x\right)=-1$

14

concept

Cases Where Limits Do Not Exist

Video duration:

3m

Video transcript

We just learned that if a function does not approach the same value from both sides, then the limit does not exist. But this is actually not the only scenario in which your limit won't exist. So here, I'm going to show you all possible cases that you may encounter in which your limit does not exist. So let's go ahead and not waste any time here and jump right into these limits. Now the first limit that we're asked to find here is the limit of f of x as x approaches 3.

Now this function might look familiar, and as we approach 3 getting really, really close from that left side, my function is going to a y value of 1. Whereas from the right side, getting really, really close to 3, it's going to a y value of 4. Because these are not the same from either side, we know that this limit does not exist. Now this won't happen for every piecewise function, but it does happen commonly for piecewise functions that have a jump like we see here. We have a jump in between one piece transitioning to the next one.

Now this is not the only case in which your limit won't exist, so let's go ahead and move on to our next limit here. Here we want to find the limit of f of x as x approaches 2. Now as we get really, really close to 2 from either side, my function appears to be going to infinity. Now if both sides are going to infinity as x approaches 2, wouldn't that just be my limit? Well, no.

Here the limit does not exist because anytime that we have unbounded behavior, meaning that our function is going to infinity or negative infinity as x approaches c, our limit does not exist because infinity is not a number. So this happens commonly for rational functions that have an asymptote. And here we do have an asymptote right at x = 2. So this is causing our unbounded behavior off to infinity, meaning that our limit does not exist. Now as you continue on in math, you may informally state the limit is equal to positive or negative infinity, but this is just used to describe the behavior of a function.

And for our purposes here anytime you see unbounded behavior as x approaches c your limit does not exist. Now let's move on to finding our final limit here. Here we want to find the limit of f of x as x approaches 0. Now looking at our graph as we get really, really close to 0 from either side, our function appears to be doing something kind of crazy. It appears to be going up and down repeatedly.

It's oscillating. Now something that you may be thinking here is couldn't I just zoom in really, really far to my graph to see exactly what it's doing as x approaches 0? And the answer is no because the further and further that we zoom in here, the more and more you'll see this function oscillating. So here, for the limit of this function as x approaches 0, the limit actually does not exist. And this will be true of any function that is oscillating near c like we see here.

Now, this will happen often for functions that look exactly like this one, the sine of 1 over x or the cosine of 1 over x. So anytime you see a function like that, watch out for a limit that does not exist. Now that we know all the cases for which limits do not exist, let's continue practicing. Thanks for watching, and I'll see you in the next one.

15

example

Cases Where Limits Do Not Exist Example 2

Video duration:

3m

Video transcript

Now that we know all of the cases for which a limit might not exist, let's go ahead and work through this example together. Feel free to try this on your own and then check back in with me. In this problem, we're asked to use the graph to estimate the value of each limit or otherwise state that it does not exist. The first limit we're asked to find is the limit of f(x) as x approaches negative 2. Observing our graph, as x approaches negative 2 from the left, I observe that my function is approaching a value of 0.

Similarly, coming in from the right and approaching negative 2, the function also seems to approach a value of 0. Thus, lim→x=−2f(x)=0. Despite the presence of a piecewise function where at x equals negative 3 something distinct occurs, this does not impact our observed limit values close to negative 2. Hence the limit equals 0, and we need not worry about other regions of the graph nearby.

Moving to the next limit, we look to find the limit of f(x) as x approaches 0. Observing the function from both the left and right as x approaches 0, it appears that the function approaches a value of 1: lim→x=0f(x)=1. This is irrespective of there being a hole in our graph at x equals 0 where the function value is negative 2, which does not affect the limit.

Finally, the limit of f(x) as x approaches negative 3 shows more complex behavior. From the left, the function seems to approach 1, but from the right, the function shows unbounded behavior descending towards negative infinity. This discrepancy between the left and right-sided limits signifies that lim→x=−3f(x) does not exist. Moreover, the unbounded behavior on the right side affirms the nonexistence of this limit.

If you have any questions about these explanations, feel free to ask. Let's continue on with some practice problems. I'll see you in the next video.

16

Problem

Problem

Use the graph of $f\left(x\right)$ to estimate the value of the limit or state that it does not exist (DNE).
$\lim_{x\rarr1}f\left(x\right)$

A

$0$

B

$1$

C

$2$

D

$DNE$

17

Problem

Problem

Use the graph of $f\left(x\right)$ to estimate the value of the limit or state that it does not exist (DNE).
$\lim_{x\rarr-2}f\left(x\right)$

A

$-3$

B

$1$

C

$4$

D

$D N E$

18

Problem

Problem

Use the graph of $f\left(x\right)$ to estimate the value of the limit or state that it does not exist (DNE).
$\lim_{x\rarr0}f\left(x\right)$

A

$0$

B

$1$

C

$5$

D

$DNE$

19

Problem

Problem

Use the graph of $f\left(x\right)$ to estimate the value of the limit or state that it does not exist (DNE).
$\lim_{x\rarr0}f\left(x\right)$

A

$-1$

B

$0$

C

$1$

D

$DNE$

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