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# Introduction to limits at infinity

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.3 (EK)
,
LIM‑2.D.4 (EK)

## Video transcript

we now have a lot of experience taking limits of functions if I'm taking the limit of f of X what we're gonna think about what is FX approach as X approaches some value a and this would be equal to some limit now everything we've done up till now is where a is a finite value but when you look at the graph of the function f right over here you see something interesting happens as X gets larger and larger it looks like our function f is getting closer and closer to 2 it looks like we have a horizontal asymptote at y equals 2 similarly as X gets more and more negative it also seems like we have a horizontal asymptote at y equals 2 so is there some type of notation we can use to think about what is the graph approaching as X gets much larger or as X gets a smaller and smaller and the answer there is limits at infinity so if we want to think about what is this graph what is this function approaching as X gets larger and larger we can think about the limit of f of X as X approaches positive infinity so that's the notation and I'm not going to give you the formal definition of this right now there in future videos we might do that but it's this idea as X gets larger and larger and larger does it look like that our function is approaching some finite value that we have a horizontal asymptote there and in this situation it looks like it is it looks like it's approaching the value 2 and for this particular function the limit of f of X as X approaches negative infinity also looks like it is approaching 2 this is not always going to be the same you could have a situation maybe we had you could have another function so let me draw a little horizontal asymptote right over here you could imagine a function that looks like this so I'm gonna do it like that and maybe does something wacky like this and it comes down and it does something like this here our limit as X approaches infinity is still 2 but our limit as X negative infinity right over here would be negative - and of course there's many situations where as you approach infinity or negative infinity you aren't actually approaching some finite value you don't have a horizontal asymptote but the whole point of this video is just to make you familiar with this notation and limits at infinity or you could say limits at negative infinity they have a different formal definition than some of the limits that we've looked at in the past where we are approaching a finite value but intuitively they make sense that these are indeed limits
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