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
My Life As an Adventure Filmmaker and Photographer (Part 1) | Nat Geo Live
I was just down in Antarctica on a really incredible expedition. We’re doing a climate change story on the wildlife and the conditions, and, uh, a fishing story as well on what’s happening down in Antarctica. The last 5 days of the journey, we crossed th…
What Basic Game Theory Teaches Us About Startups
They never get the lessons in little dabs along the way. Like, you know, as kids, we’re used to getting these little lessons along the way. For these zero-sum games, often the lesson just comes fast and hard at the end. It’s like, “Oh!” This is Michael Se…
4 Reasons to Invest NOW | Ask Mr Wonderful Shark Tank's Kevin O'Leary
Sometimes the entire year’s return comes in just a few days, and if you’re not invested in those days, you miss out. Your 20s and 30s are prime earning years. The longer you wait, the less you will have in retirement, so it’s best you get started right no…
Polynomial identities introduction | Algebra 2 | Khan Academy
What we’re going to do in this video is talk a little bit about polynomial identities, and this is really just a fancy way of seeing whether an expression that involves a polynomial is equal to another expression. So, for example, you’re familiar with x …
The Most Powerful Mindset for Success
There is a psychological trait that all successful people appear to have in common. It’s been cosigned by Bill Gates and NASA uses it as a criteria for selecting potential Systems Engineers. This concept is called the growth mindset, a term originally coi…
Lion Falls From Tree During Rescue From a Wire Snare Injury | National Geographic
We pretty much picked up on finding that Orinda at a point where she was treated about two weeks back. So even though the snare was removed, we saw that over about a two and a half week period, her conditions started again to rapidly decline. So we find …