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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.

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