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

Approximating limits using tables | Limits and continuity | AP Calculus AB | Khan Academy


3m read
·Nov 11, 2024

This video we're going to try to get a sense of what the limit as x approaches 3 of ( x^3 - 3x^2 ) over ( 5x - 15 ) is. And when I say get a sense, we're going to do that by seeing what values for this expression we get as x gets closer and closer to 3. Now, one thing that you might want to try out is: well, what happens to this expression when x is equal to 3?

Well then, it's going to be ( 3^3 - 3 \times 3^2 ) over ( 5 \times 3 - 15 ). So at ( x = 3 ), this expression is going to be, and see, the numerator we have ( 27 - 27 = 0 ) over ( 15 - 15 = 0 ). So this expression is actually not defined at ( x = 3 ). We get this indeterminate form; we get ( 0/0 ).

But let's see, even though the function, even though the expression is not defined, let's see if we can get a sense of what the limit might be. And to do that, I'm going to set up a table. So let me set up a table here, and actually, I'm going to set up two tables. So this is ( x ), and this is ( \frac{x^3 - 3x^2}{5x - 15} ). And actually, I'm going to do that again, and I'll tell you why in a second. So this is going to be ( x ), and this is ( \frac{x^3 - 3x^2}{5x - 15} ).

The reason why I set up two tables, I didn't have to do two tables; I could have done it all in one table. But hopefully, this will make it a little bit more intuitive. What I'm trying to do is on this left table, I'm going to let's try out x values that get closer and closer to 3 from the left, from values that are less than 3. So for example, we go to ( 2.9 ) and figure out what the expression equals when ( x ) is ( 2.9 ). But then we could try to get even a little bit closer than that; we could go to ( 2.99 ), and then we could go even closer than that; we could go to ( 2.999 ).

One way to think about it here is, as we try to figure out what this expression equals as we get closer and closer to 3, we're trying to approximate the limit from the left. So, limit from the left. And why do I say the left? Well, if you think about this on a coordinate plane, these are the x values that are to the left of 3, but we're getting closer and closer and closer, we're moving to the right. But these are the x values that are on the left side of 3; they're less than 3.

But we also, in order for the limit to exist, we have to be approaching the same thing from both sides, from both the left and the right. So we could also try to approximate the limit from the right. And so what values would those be? Well, those would be x values larger than 3. So we could say ( 3.1 ), but then we might want to get a little bit closer; we could go ( 3.01 ), but then we might want to get even closer to 3, ( 3.001 ). Every time we get closer and closer to 3, we're going to get a better approximation for—or we're going to get a better sense of what we are actually approaching.

So, let's get a calculator out and do this. And you could keep going ( 2.99999999 ), ( 3.0001 ). Now, one key idea here to point out before I even calculate what these are going to be: sometimes when people say the limit from both sides, or the limit from the left, or the limit from the right, they imagine that the limit from the left is negative values and the limit from the right are positive values. But as you can see here, the limit from the left are to the left of the x value that you're trying to find the limit at. So these aren't negative values; these are just approaching the 3 right over here from values less than 3. This is approaching the 3 from values larger than 3.

So now let's fill out this table, and I'm speeding up my work so that you don't have to sit through me typing everything into a calculator. So based on what we're seeing here, I would make the estimate that this looks like it's approaching ( 1.8 ). So is this equal to ( 1.8 )? As I said, in the future, we're going to be able to find this out exactly. But if you're not sure about this, you could try even closer and closer and closer values.

More Articles

View All
Rob Riggle Ice Climbing in Iceland | Running Wild With Bear Grylls
BEAR GRYLLS: OK, Rob. Your front points– your crampons are your main weight-bearing things. Good lord. BEAR GRYLLS (VOICEOVER): Comedian Rob Riggle and I are in a race against time, searching to find a case of supplies before nightfall. But first, we’ve …
WEIRDEST Headphone Site! ... and MORE! IMG! #27
A chicken made out of eggshells and locked and loaded. It’s episode 27 of IMG. This guy loves dogs, and this guy hates his son. B.C.A. brought us angry bat birds, complete with Robin, Alfred, the Joker, Bane, and used lipstick to make a panda Pikachu. Thi…
The Loner's Path | Philosophy for Non-Conformists
The Loner’s Path | Philosophy for Non-conformists The path of nonconformity is alluring to those who don’t seek to follow the herd known as a society. Instead, they want to make unique individual choices in life, disregarding other people’s opinions and …
Building Shelter | How to Survive on Mars
When we get to Mars, we need to solve our basic needs, in particular protection from radiation. The first crew that lands on Mars will live in their ship, but you can’t live there very long. The cosmic radiation and the solar radiation is going to penetra…
136 Countries Agree To Global Minimum Corporate Tax Rate!
Hey guys, welcome back to the channel! So in this video, we have some interesting news to me. I guess probably a lot of people would zone out at the thought of corporate tax rates, but to me, we have some interesting news. Because last Friday, 136 countr…
How I spent $50,000 in South America - Not Forgotten SED 107
Hey it’s me Destin, welcome back to Smarter Every Day. So the purpose of this particular video is to convince you to click at the end of the video on one thing that will change a child’s life. If you’re an evil person and you want to do bad things to lit…