Functions continuous at specific x-values | Limits and continuity | AP Calculus AB | Khan Academy

Which of the following functions are continuous at x = 3?

Well, as we said in the previous video, in the previous example, in order to be continuous at a point, you at least have to be defined at that point. We saw our definition of continuity: f is continuous at a if and only if the limit of f as x approaches a is equal to f of a.

So over here, in this case, we could say that a function is continuous at x = 3. So f is continuous at x = 3 if and only if the limit as x approaches three of f(x) is equal to f(3).

Now let's look at this first function right over here: natural log of(x - 3). Well, try to evaluate— and it's not f now, it's g. Try to evaluate g(3).

Let me write it here: g(3) is equal to the natural log of(3 - 3). This is not defined. You can't raise e to any power to get to zero. You try to go to, you know, you could say negative infinity, but that's not— this is not defined. And so if this isn't even defined at x equals 3, there's no way that it's going to be continuous at x = 3. So we could rule this one out.

Now, f(x) is equal to e^(x - 3). Well, this is just a shifted over version of e^x. This is defined for all real numbers, and as we saw in the previous example, it's reasonable to say it's continuous for all real numbers.

You could even do this little test here: the limit of e^(x - 3) as x approaches 3. Well, that is going to be e^(3 - 3) or e^0, or 1. And so f is the only one that is continuous.

And once again, it's good to think about what's going on here visually. If you like, both of these— you could think of them as a shifted over version of ln(x). This is a shifted over version of e^x.

And so, if we like, we could draw ourselves some axes. So that's our y-axis; this is our x-axis. And actually, let me draw some points here.

So that's 1, that is 1, that is— let's see— 2, 3, 2 and 3. And let's see, these are— I said these are shifted over versions so actually this is maybe not the best way to draw it. So let me draw it this is 1, 2, 3, 4, 5, and 6. And on this axis, I won't make them with the same scale. Let's say this is 1, 2, 3.

I'm going to draw a dotted line right over here. So g(x) = ln(x - 3) is going to look something like this— this, if you put three in it, it's not defined. If you put four in it, ln(4) well that's— oh sorry— ln(4) - 3 is— actually let me just draw a table here. I know I'm confusing you.

So if I say x and I say g(x), so at three, undefined; at four, this is ln(1), ln(1) which is equal to 0. So it's right over there. So g(x) is going to look something like that. And so you can see at three, it's— you have this discontinuity there. It's not even defined to the left of three.

Now f(x) is a little bit more straightforward. If you have— so e^3 is going to be— or sorry, f(3) is going to be e^(3 - 3) or e^0, so it's going to be 1. So it's going to look something like this— it's going to look something— something like that.

There's no jumps, there's no gaps; it is going to be continuous at frankly all real numbers. So for sure it's going to be continuous at three.