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

Limits are fundamental in calculus, representing the y-value a function approaches as x nears a specific value from either side. This can be analyzed graphically or numerically. One-sided limits, denoted with negative or positive signs, help determine if a limit exists. If the left and right limits differ, the limit does not exist (DNE). Additionally, limits can fail to exist due to unbounded behavior or oscillation near a point. Understanding these concepts is crucial for analyzing functions and their behaviors in calculus.

Finding Limits Numerically and Graphically

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.

Now 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 x2 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 right 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 x2 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.

Find the limit by creating a table of values.

$\lim_{x\rarr0}-4x+2$

• A. $-4$
• B. $-2$
• C. $0$
• D. $2$

Find the limit by creating a table of values.

$\lim_{x\rarr2}3x^2+5x+1$

• A. $1$
• B. $10$
• C. $23$
• D. $21$

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. $2$

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

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

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

Finding Limits Numerically and Graphically Example 1

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 approaches negative 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. And 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 approaches negative 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 approaches 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 approaches 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 approaches 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 approaches 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 coming up next in our next video. So I'll see you there. Thanks for watching, and let me know if you have questions.

One-Sided Limits

When finding a limit, we look for the y value that our function is going to as x gets really, 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 explicitly asked 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 the notation. So let's go ahead and jump right in.

In working with a one-sided limit, we still write the limit of f(x), our function. But instead of x just approaching c, we have x approaching c with a little negative sign. And we read this as the limit of f(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(x) as x approaches c with a little positive sign from the right. To remember which sign represents which limit, 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. Now that we've seen this notation, let's take a look at this function here and actually find some limits.

Let's start here with our left-sided limit. So, the limit of f(x) as x approaches 3 from the left. 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. I can plug values getting really, really close to 3 from the left side into my function. Here as we're closing in on 3, I use values like 2.992, 2.999. Now, because this is a piecewise function and these values are less than 3, my function here is equal to x minus 2. So plugging these values into my function, I get outputs like 0.992, 0.999. Just like I saw on my graph, based on these values, my function is going to a y value of 1. So here, my left-sided limit, the limit of f(x) as x approaches 3 from the left, is 1.

Now, let's take a look at our right-sided limit here, the limit of f(x) as x approaches 3 from the right. Looking at our graph, 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. I look at those values getting really, really close to 3 from that right side, at 3.001, 3.013. Again, plugging those into my function, which here since these values are greater than or equal to 3, my function is equal to 4. Based on these values, just like I saw on my graph, I see that my function is going to a y value of 4. So, my right-sided limit, the limit of f(x) as x approaches 3 from the right, is equal to 4.

Now, what if I asked you to find the limit of f(x) as x approaches 3? Not from the left or from the right, but just as x approaches 3. This is a bit confusing because our one-sided limits show different outcomes. As x gets really, really close to 3 from either side, we see different things happening. Because this function does not approach the same value from both sides, the limit does not exist or is abbreviated as DNE. Here, with this function, as x approaches 3 from either side, going to a y value of 1 from one side and to a y value of 4 from the other, the limit does not exist. Here we were able to find our one-sided limits, and because these one-sided limits were not equal, that told us that our limit does not exist.

Let's take a look at another example. We use this graph to find the given limits. Now, the first limit we're asked to find here is the limit of f(x) as x approaches 1 from the left. As x approaches 1 coming in from that left side, I see that my function is going to a y value of -1. So here, the limit of f(x) as x approaches 1 from the left, my left-sided limit, is -1.

Now, moving on to our next limit. Here we have our right-sided limit, the limit of f(x) as x approaches 1 from the right. Again, taking a look at our graph, as x gets really close to 1 coming in from that right side, I see again that my function is going to a y value of -1. So here, my right-sided limit is the same as my left-sided limit, -1.

Now, let's look at one final limit, the limit of f(x) as x approaches 1. We want to pay attention to what's happening as x approaches 1 from either side, but we have already done that by finding our one-sided limits. Here, since our one-sided limits are equal, they are both -1, that tells us that our limit here, the limit of f(x) as x approaches 1, is also -1. 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.

Now that we've seen and put a notation to these one-sided limits and know how to use them to determine whether a limit exists, let's continue practicing. Thanks for watching, and I'll see you in the next one.

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. $\lim_{x\rarr-2^{-}}f\left(x\right)=1$, $\lim_{x\rarr-2^{+}}f\left(x\right)=1$, $\lim_{x\rarr-2^{}}f\left(x\right)=1$
• B. $\lim_{x\rarr-2^{-}}f\left(x\right)=1$, $\lim_{x\rarr-2^{+}}f\left(x\right)=-1$, $\lim_{x\rarr-2}f\left(x\right)=DNE$
• C. $\lim_{x\rarr-2^{-}}f\left(x\right)=1$, $\lim_{x\rarr-2^{+}}f\left(x\right)=1$ , $\lim_{x\rarr-2}f\left(x\right)=DNE$
• D. $\lim_{x\rarr-2^{-}}f\left(x\right)=0$, $\lim_{x\rarr-2^{+}}f\left(x\right)=0$, $\lim_{x\rarr-2}f\left(x\right)=0$

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

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

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

Cases Where Limits Do Not Exist

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(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(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 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(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.

Cases Where Limits Do Not Exist Example 2

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. Now here 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.

Now the first limit that we're asked to find here is the limit of \( f(x) \) as \( x \) approaches negative 2. So looking at our graph here, my \( x \) value of negative 2 coming in from that left side, getting really, really close to negative 2, I see that my function seems to be approaching a value of 0. Then coming in from that right side again getting really, really close to negative 2, I see again that my function seems to be approaching a value of 0. So the limit of \( f(x) \) as \( x \) approaches negative 2 is equal to 0. Now, something that might trip you up here is that we have sort of a piecewise function going on. So once \( x \) is equal to negative 3, something completely different is happening. So you might be tempted to say, okay, well, as \( x \) is approaching negative 2, it's doing something crazy over here. So this limit does not exist. But you would be wrong in saying that because remember that when looking at limits, we are looking at values really, really close to whatever \( x \) value we're looking at. So negative 3 and negative 2, even though you might think of these values as being pretty close together, remember that we're looking at values really, really close to negative 2, like negative 2.01 or negative 2.001. So here, our limit is equal to 0. We don't have to worry about what else is going on with our graph over there. Now let's move on to our next limit.

Here, we're asked to find the limit of \( f(x) \) as \( x \) approaches 0. So on our graph here, as \( x \) approaches 0 from that left side, I see my function is approaching a value of 1. Then coming in from that right side, I see the same thing happening. As \( x \) gets really, really close to 0, my function is approaching a value of 1. So here, my limit of \( f(x) \) as \( x \) approaches 0 is 1. Now, something else that might have tripped you up here is that we have this hole in our graph. So the actual value of our function when \( x \) is equal to 0 is actually negative 2. But remember that this does not matter because when looking at our limit, we are looking at what our function is doing as \( x \) gets really, really close to 0, not when \( x \) is 0. So our function or our limit here is 1, and we can move on to our next limit.

Now the final limit that we're asked to find here is the limit of \( f(x) \) as \( x \) approaches negative 3. Now looking at our graph here, I can already see that something a little crazy is going on here around \( x \) equals negative 3, but let's dig a bit deeper here. Now looking at \( x \) approaching negative 3 from that left side, I see that my function is approaching a value of 1. Now from that right side, it is definitely not doing that. And I see that as we get really, really close to negative 3 coming in from the right, then my function is actually experiencing some unbounded behavior. And my function is going to negative infinity. Now remember that here when we're dealing with unbounded behavior, our limit does not exist. Now that's just our right sided limit. But we have another problem here because from the left and the right, our function's not doing the same thing. So we already know that our limit of \( f(x) \) as \( x \) approaches negative 3 simply does not exist because our one-sided limits are not the same. And one of our one-sided limits is actually already doesn't exist because it's experiencing unbounded behavior going to negative infinity.

Now let me know if you had any questions here, but let's continue on with some practice problems. I'll see you in the next video.

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

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. DNE

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

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. $-1$
• B. $0$
• C. $1$
• D. DNE

Here’s what students ask on this topic:

What is the definition of a limit in calculus?

A limit in calculus is the value that a function (f(x)) approaches as the input (x) approaches a certain value (c). Formally, the limit of f(x) as x approaches c is denoted as:

$lim{x}_{\to c}f\left(x\right)$

This means that as x gets arbitrarily close to c from either side, the function f(x) gets arbitrarily close to a specific value L. Limits are fundamental in understanding the behavior of functions at specific points, especially where the function may not be explicitly defined.

How do you find the limit of a function using a graph?

To find the limit of a function using a graph, follow these steps:

1. Identify the point c at which you want to find the limit.
2. Observe the behavior of the function as x approaches c from the left (x → c^-) and from the right (x → c⁺).
3. If the function approaches the same y-value from both sides, that y-value is the limit.
4. If the function approaches different y-values from each side, the limit does not exist (DNE).

For example, if you want to find the limit of f(x) as x approaches 2, look at the y-values the function approaches as x gets close to 2 from both sides. If both sides approach the same y-value, that is your limit.

What are one-sided limits and how are they different from regular limits?

One-sided limits consider the behavior of a function as x approaches a specific value from only one side—either from the left or the right. They are denoted as:

Left-sided limit: $lim{x}_{\to c-}f\left(x\right)$

Right-sided limit: $lim{x}_{\to c+}f\left(x\right)$

Regular limits require that the function approaches the same value from both sides. If the left-sided and right-sided limits are equal, the regular limit exists and is equal to that value. If they differ, the regular limit does not exist (DNE).

What are the scenarios in which a limit does not exist?

A limit does not exist (DNE) in the following scenarios:

1. Different One-Sided Limits: If the left-sided limit and right-sided limit as x approaches c are not equal.
2. Unbounded Behavior: If the function approaches infinity or negative infinity as x approaches c. For example, rational functions with vertical asymptotes.
3. Oscillating Behavior: If the function oscillates between values as x approaches c, such as $\sin \left(\frac{1}{x}\right)$ or $\cos \left(\frac{1}{x}\right)$.

In these cases, the function does not approach a single finite value as x approaches c, so the limit does not exist.

How do you find the limit of a function using a table of values?

To find the limit of a function using a table of values, follow these steps:

1. Choose values of x that get increasingly close to the point c from both the left and the right.
2. Calculate the corresponding f(x) values for these x values.
3. Observe the trend of the f(x) values as x approaches c.
4. If the f(x) values approach the same number from both sides, that number is the limit.

For example, to find the limit of f(x) as x approaches 2, you might use x values like 1.9, 1.99, 1.999 (from the left) and 2.1, 2.01, 2.001 (from the right). If the f(x) values approach the same number, that is your limit.