Worked example: Continuity at a point | Limits and continuity | AP Calculus AB | Khan Academy

We have the graph of y is equal to g of x right over here. What I want to do is check which of these statements are actually true and then check them off. Like always, I encourage you to pause the video and see if you can work through this on your own.

Let's look at this first statement. This first statement says both the limit of g of x as x approaches 6 from the right-hand side and the limit as x approaches 6 from the left-hand side of g of x exist.

Alright, so let's first think about the limit of g of x as x approaches 6 from the right-hand side, as we approach 6 from values greater than 6. If we look over here, we could say okay, when x is equal to 9, g of 9 is right over there. g of 8 is right over here, g of 7 is right over here. It looks like it's between negative 3 and negative 4.

g of 6.5 looks like it's a little bit... it's a little... it's still between negative 3 and negative 4, it's but it's closer to negative 3. g of 6.1 is even closer to negative 3. g of 6.01 is even closer to negative 3. So it looks like the limit from the right-hand side does exist. So it looks like this one exists.

Now let's see, and I'm just looking at it graphically, and that's all they can expect you to do in an exercise like this. Now let's think about the limit as x approaches 6 from the left-hand side. I could start anywhere, but let's say when x is equal to 3, g of 3 is a little more than one.

g of 4 is looks like it's a little bit less than 2. g of 5 looks like it's close to 3. g of 5.5 looks like it's between 5 and 6. g of 5.75 looks like it's approaching 9. As we get closer and closer, as x gets closer and closer to 6 from below, from values to the left of 6, it looks like we're unbounded, we are approaching infinity.

So technically, we would say this limit does not exist, so this one does not exist. I won't check this one off. Some people will say the limit is approaching infinity, but technically, infinity is not a value that you can say it is approaching in the classical formal definition of a limit. So for these purposes, we would just say this does not exist.

Now, let's see. They say the limit as x approaches 6 of g of x exists. Well, the only way that the limit exists is if both the left and the right limits exist and they approach the same thing. Well, we don't even... our left limit, our limit as x approaches 6 from the negative side or from the left-hand side, I guess I could say, does not even exist. So this cannot be true.

So that's not going to be true: g is defined at x equals 6. At x equals 6, it doesn't look like g is defined. Looking at this graph, I can't tell you what g of 6 should be. We have an open circle over here, so g of 6 is not equal to negative 3.

And this goes up to infinity, and we have a vertical asymptote actually drawn right over here at x equals six, so g is not defined at x equals six. So I'll rule that one out.

g is continuous at x equals six. Well, you can see that it goes up to infinity, then it jumps down back down here, then continues. So just when you just think about it in common sense language, it looks very discontinuous.

If you want to think about it more formally, in order for something to be continuous, the limit needs to exist at that value, the function needs to be defined at that value, and the value of the function needs to be equal to the value of the limit. Neither of these, the first two conditions, aren't true, and so these can't even equal each other because neither of these exist.

So this is not continuous at x equals six. The only thing I could check here is none of the above. Let's do another one of these. So the first statement: both the right-hand and the left-hand limit exist as x approaches 3.

So let's think about it. So x equals 3 is where we have this little discontinuity here, this jump discontinuity. So let's approach, let's go from the positive, from values larger than three. So when x is equal to five, g of five is a little bit more negative than negative three.

g of four is between negative two and negative three. g of 3.5 is getting a little closer to negative 2. g of 3.1 is getting even closer to negative 2. g of 3.01 is even closer to negative 2. So it looks like this limit right over here... oh, I'm circling the wrong one. It looks like this limit exists, and in fact, it looks like it is approaching negative 2.

So this right over here is equal to negative 2, the limit of g of x as x approaches 3 from the right-hand side. I'll think about it from the left-hand side. So I can start here: g of 1 looks like it's a little bit greater than negative one.

g of two, it’s less than one. g of two point five is between one and two. g of two point nine looks like it's a little bit less than two. g of 2.99 is getting even closer to 2. g of 2.99999 would be even closer to... so it looks like this thing right over here is approaching 2.

Both of these limits, the limit from the right and the limit from the left exist; the limit of g of x as x approaches 3 exists. So these are the one-sided limits. This is the actual limit.

Now, in order for this to exist, both the right and left-handed limits need to exist, and they need to approach the same value. Well, this first statement, we saw that both of these exist, but they aren't approaching the same value. From the left, we are approaching 2.

So this limit does not exist, so I will not check that out or I will not check that box. g is defined at x equals 3. When x equals 3, we see a solid dot right over there, and so it is indeed defined.

It is indeed defined there. g is continuous at x equals 3. Well, in order for g to be continuous at x equals 3, the limit must exist there, it must be defined there, and the value of the function there needs to be equal to the value of the limit.

Well, the function is defined there, but the limit doesn't exist there, so it cannot be continuous. It cannot be continuous there. So I would cross that out, and I can't click... I can't... I wouldn't click none of the above because I've already checked something right; I've actually checked two things already.