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

Graphical limit at asymptotic discontinuity


2m read
·Nov 11, 2024

All right, we have a graph of ( y ) is equal to ( f(x) ), and we want to figure out what is the limit of ( f(x) ) as ( x ) approaches negative three. If we just look at ( x = -3 ), it's really hard to see, at least based on how this graph looks, what ( f(-3) ) is. If anything, it looks like we have an asymptotic discontinuity here.

It looks like, on the left side of ( x = -3 ), we're approaching, I guess you could say, infinity; and on the right side, it looks like we're approaching infinity as well. We could just look at that and say, "Well, look, what is ( f(-5) )?" Well, it's 4. ( f(-4) ) looks like it's around, I don't know, around 8. ( f(-3) ) is off the charts. If we continued with this trend, and if we were to asymptote towards this line right over here, this vertical asymptote, it looks like as we get closer and closer to negative 3, the value of the function at that point is approaching—it's getting closer and closer to infinity.

At least that's what it looks like from what we can see on this graph as we approach negative 3 from the left-hand side. Let's think about what's happening as we approach negative 3 from the right-hand side. So this is ( f(-1) ), ( f(-2) ), and ( f(-2.5) ) looks like it's up here someplace. ( f(-2.9) ) would be even higher, and ( f(-2.999) ) looks like it would just once again approach infinity.

So this type of limit, in some context, you would say that this limit doesn't exist, doesn't exist in the formal sense. So that's one way to think about it. In some contexts, you will hear people say that this limit, since from the left and from the right, it looks like it's going to infinity. Sometimes you will see people say that it is approaching infinity, and so this is depending on what type of class or context you're in. But in the traditional sense of the limit, or in the technical sense, there are ways that you can define limits where this would make a little bit more sense. However, the traditional definition of a limit would be you would say that this limit does not exist.

More Articles

View All
A Submarine Assault | WW2 Hell Under the Sea
July 31st, 1944. With Commander Lawson Ramage fixated on another target in Japanese convoy MI-11, below deck, battle helmsman Chet Stanton has made the decision to evade an escort that threatens to ram the American submarine. The crew of USS Parche wait t…
National Geographic goes to space with Disney and Pixar's Lightyear
(Heroic music) I’m Nadia Drake with National Geographic. Today, we’re at NASA’s Jet Propulsion Laboratory with Chris Evans, who plays Buzz Lightyear in the new Disney and Pixar movie Lightyear. Hanging out with the intrepid freshly returned from orbit, as…
How The Internet Changed Everything
[Music] In August 1962, JCR Licklider proposed a new but monumental idea: computers that could talk to one another. A simple idea, but one whose implications resulted in a world-changing network. The first message sent over the Internet, which at this tim…
Trick involving Maclaurin expansion of cosx
The first three nonzero terms of the McLaurin series for the function ( f(x) = x \cos(x) ). So one thing that you’re immediately going to find, let’s just remind ourselves what a McLaurin series looks like. Our ( f(x) ) can be approximated by the polynom…
The Technical Advisor for Silicon Valley on HBO: Ed McManus
Okay, so today we have Ed McManis. He was a technical adviser for Silicon Valley, uh, on HBO season 3. Um, so Ed, what’s your background? Okay, so, uh, I was a technical co-founder of a Y Combinator startup called Yard Sale. Um, and, uh, we launched two …
2010 Berkshire Hathaway Annual Meeting (Full Version)
[Applause] Good morning. I’m Warren, he’s Charlie. He can hear, I can see. We work together for that reason. I’d like to make one correction: in the movie, my fastball was filmed in slow motion. They tried it the regular way, and you couldn’t even see it.…