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## Limits at infinity (horizontal asymptotes)

Current time:0:00Total duration:7:41

# Infinite limits intro

## Video transcript

Let's say that f of x
is equal to 1 over x. And we want to think about
what the limit of f of x is as x approaches 0 from
the positive direction. And to think about
this I'm going to set up a little table here. So let's set up x and then
let's think about what f of x is going to be. I'm going to approach 0
from the positive direction. So let's say we try 0.1. Then we're going to try 0.01. Then we can try 0.001. Then we can try 0.0001. So notice-- each
of these numbers-- they're all larger than 0
and they're approaching 0 from the positive direction. We're getting closer and
closer and closer to 0. So when x is 0.1, f of x is
just going to be 1 over this. This is 1/10 so 1 over that
is just going to be 10. To 1 over 0.01 is
going to be 100. 1 over 0.001 is
going to be 1,000. 1 over 0.0001 is
going to be 10,000. So you see, as x gets
closer and closer to 0 from the positive
direction, f of x just really grows
really, really fast. So what we say here
is the limit of f of x as x approaches 0
from the positive direction is going to be equal
to positive infinity. Or we could just write infinity. This thing over here-- if
we put something really, really close, so if
we say 0 point seven digits behind the decimal
place, then 1 over that's going to be 1 with one,
two, three, four, five, six, seven zeros. Did I do that right? Here I had four places
behind the decimal, four 0's. Here I have one, two, three,
four, five, six, seven, and here I have seven 0's. So you see, as we get
closer and closer to 0 from the positive
direction, the f of x just gets larger and
larger and larger. It's just completely unbounded. So we'd say this is
equal to infinity. Well, let's think
about another limit. Let's think about the
limit as x approaches 0 from the negative direction
of f of x, or the limit of f of x as x approaches 0 from
the negative direction. Well in that case, we can
just make each of these values negative. So if x is negative
0.1 then this is going to be negative 10. If this is negative,
then this is negative. If this is negative,
then this is negative. If this is negative,
then this is negative. If this is negative,
then this is negative. And so what we see here is
that this gets more and more-- becomes larger
and larger numbers in the negative direction. If we keep going, if we're
thinking about a number line, further and further and
further to the left. So we can say the
limit of f of x as x approaches 0 from
the negative direction is equal to negative infinity. Well that's interesting. Now let's think about
a limit as x approaches either positive or
negative infinity. So let's now think
about the limit of f of x as x approaches infinity. And one way to set up this
table-- we can just say-- do a similar thing. x and f of x-- so if x is
10, then f of x is 1 over 10. If x is-- and I'm just going to
go larger and larger numbers-- if x is 1,000, then f
of x is 1 over 1,000. If x is 1,000,000, then f of
x is going to be 1/1,000,000. So you see as x gets larger
and larger and larger in the positive direction,
this f of x now gets closer and closer
and closer to 0. So we can say the
limit of f of x as x approaches
infinity is equal to 0. Now let's think
about the limit of f of x as x approaches
negative infinity. So we're going to
take numbers that are more and more
and more negative. Well if x is negative 10, this
is going to be negative 1/10. If x is negative 1,000, this is
going to be negative 1/1,000. If this is negative 1,000,000,
this is going to be negative 1/1,000,000. But we still see that
we are approaching 0. So here, once again, we are
once again approaching 0. So what implications
does this have, besides that we've just been
able to deal with limits. And once again, I
haven't given you a formal definition of this,
but it's hopefully giving you an intuition as we
take limits to infinity to negative infinity--
actually this is supposed to be negative infinity--
limits to infinity, limits to negative infinity, or when
our limited self is infinity, or negative infinity. So one, we're seeing
that we can do that. But let's actually
try to visualize this when we look at the graph
of f of x is equal to 1 over x. So let's do it--
actually maybe I want to be able to keep
looking at all of this stuff, so let me set up the graph. right over here. So that's our x-axis. This right over
here is our y-axis. And let's graph f of x. So we see that if x is a very
small number, if x is 0.1, then y-- y is equal
to f of x-- is going to be a very high number. And as the closer and
closer we get to 0 from the positive direction,
f of x approaches infinity. So it just keeps
approaching infinity as we get closer
and closer to 0. As x gets closer
and closer to 0, the y value just gets
higher and higher. Then, as our x value gets larger
and larger, our f of x value gets smaller and smaller. So it looks something
like that it approaches 0. Similarly, if we approach x
from the negative direction right over here,
we saw that f of x is approaching
negative infinity. So as we get x is closer
and closer to 0, our f of x gets more and more
and more negative. And then as our x becomes
more and more negative, the x itself becomes
more and more negative. We see that our function
is approaching 0. So the way I've drawn it,
we see that there's actually a two asymptotes for the graph
of f of x is equal to 1/x. You have a horizontal
asymptote at y is equal to 0. When x approaches infinity, f
of x gets closer and closer to 0 but never quite touches it. When x approaches
negative infinity, f of x is getting closer and
closer to 0 from the bottom but it never quite touches it. And we also have a vertical
asymptote right over here at x is equal to 0. And we see that because
as x approaches 0 from the positive direction,
y approaches infinity. And as x approaches 0 from
the negative direction, y approaches negative infinity. So the limit here, at x is equal
to 0-- so if you were to say, we looked at the limit
as x approaches 0 from the positive direction and
from the negative direction, but we see that
they're approaching two different things. So we definitely have a vertical
asymptote at x is equal to 0. But the limit as x
approaches 0 of f of x-- this is not defined. Why is that? Well, when we approach 0
from the positive direction we get a different thing
than when we approach it from the negative direction.