yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


3m read
·Nov 11, 2024

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.

More Articles

View All
Ballet Shoes: The Craft Before the Dance | Short Film Showcase
My name’s Tony Collins of Jim to free data. As a young lad, I arrived on the end of ‘69, 1969. I’ve been here ever since and out the lot of it at the old school from where we originated from Leicester Square. This is about three of us left in the firm sti…
TikTok Is Causing A Mass Psychosis
[Music] In June 2019, Kirsten Muellerval, a psychiatrist at Hanover Medical School and head of its Tourette’s outpatient department, noticed unusual symptoms in her new set of patients. To begin with, all of them were teenagers, and they were suffering fr…
Swimming With Sharks: Photographing the Ocean’s Top Predators (Part 1) | Nat Geo Live
What I’d like to share with you this evening is some of my latest work for National Geographic about sharks. Or, as we say where I come from in Massachusetts, sharks. Over the last two years, I’ve worked on four separate projects. Four separate stories ab…
Meme Culture: How Memes Took Over The World
Ah, here we go again. On the 1st of September 1939, Germany invaded Poland from the east, starting World War II. As you would expect, there is fear and panic throughout Europe. So, to calm the British population down and to prevent widespread panic, the w…
Morgan DeBaun on Reaching 20M Millennials - With Kat Manalac at the Female Founders Conference
And now I’m really, really excited to introduce you to our next speaker, Morgan DeBon. She’s the founder of Blabbetty. So, Blabbetty has, you know, grown into the largest media company and lifestyle brand for Black Millennials. Morgan started Blabbetty in…
5 Habits That Made Me $30,000/month By 22
I went from making seven dollars per hour at the restaurant to earning multiple six figures in only two and a half years from the YouTube businesses I’ve built. Growing up, unfortunately, I didn’t receive much financial education, and as everyone knows, s…