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## Calculus, all content (2017 edition)

### Unit 1: Lesson 20

Limits at infinity (horizontal asymptotes)- Infinite limits intro
- Limits at infinity of quotients (Part 1)
- Limits at infinity of quotients (Part 2)
- Limits at infinity of quotients
- Limits at infinity of quotients with square roots (odd power)
- Limits at infinity of quotients with square roots (even power)
- Limits at infinity of quotients with square roots
- Limits at infinity of quotients with trig
- Limits at infinity of quotients with trig (limit undefined)
- Limits at infinity of quotients with trig
- Limit at infinity of a difference of functions

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# Infinite limits intro

Here we consider the limit of the function f(x)=1/x as x approaches 0, and as x approaches infinity. Created by Sal Khan.

## Want to join the conversation?

- is there a way f(x) = 0, if infinity isn't a number?(25 votes)
- There is no finite number that you can plug into 1/x to give zero. If you think about it as an algebraic expression ( y = 1/x ), plug 0 in for y and try to solve for x.

0 = 1/x (multiply both sides by x)

0*x = 1

We know that this cannot happen, as 0 times anything is 0. Therefore this is equation has no solution.(166 votes)

- Is there a difference between a limit that does not exist and a limit that is undefined? If so can someone elaborate :)(46 votes)
- Good question! A limit that does not exist and a limit that is undefined mean exactly the same thing. Just like the slope of a vertical line: It does not exist, and it is undefined.(47 votes)

- One question that is bugging me:

let be c the value that x approaches to, where c is within [-infinity, +infinity] and x is a Real number. According how Real numbers are defined, there is no real number x >= +infinity. After Khans explanation, in order a limit is defined, the following predicate must be true: if and only if lim x->c f(x), then lim x->c+ f(x) = lim x->c- f(x). But since there is no x where x >= +infinity, a limit where x approaches to infinity is undefined.

In other words: There is no real number x, that can approach to infinity from both sides, therefore a limit where x approaches to infinity is undefined.

I hope someone can enlighten me.(4 votes)- Great question!!

A HUMAN once said:

“...it's very much like your trying to reach infinity. You know that it's there, you just don't know where-but just because you can never reach it doesn't mean that it's not worth looking for.”

― Norton Juster, The Phantom Tollbooth

So the point is beautiful you cannot show infinity on a graph, but you can at least talk about it and show it by some sort of an identifier, an image, or a symbol.

Its almost like: nobody can tell you what apple tastes like, you have to be given the apple to find out but at least the people who have eaten an apple in their lives can talk among themselves and still know what they are talking about!

- Vedanta, Indian religious(véros science) text

I hope that was of help!!(1 vote)

- How can their be a infinity on both sides of the number line? If you have a large number on the positive side, their is two times that number from the amount of numbers from the positive value and the negative value, right? If so, does this mean that their are different sizes of infinity? How is that possible, if it is true, when infinity goes on forever?(5 votes)
- Infinity can be mathematically defined as an unbounded quantity greater than every real number. (http://www.wolframalpha.com/input/?i=infinity) To picture that on a number line, it is the quantity or number that is greater than anything we can imagine. The number line just keep going and going, greater and greater. Now, if that is in the positive direction on the number line, what about in the negative direction? I can imagine a very negative number (or a negative number with a great absolute value) such as -1,000,000,000 but there is always a quantity that will be more negative, hence "negative infinity."

I would not say infinity has a "size," because once you quantify how big it is, there can be something bigger. Think of positive and negative infinity as describing the direction of the unbounded value. Say we have a light source at 0 on the number line. One photon travels in the positive direction. Another photon travels in the negative direction. How far will the photons travel? If we leave the time frame undefined, the photons will travel an infinite distance, but in opposite directions.

Does this answer help you understand infinity, Humza?(3 votes)

- This is a super long question, so please bear with me:

At2:59, Sal guides us to "think about a limit as x approaches either positive or negative infinity." Similarly in my Calculus classes, I have been taught that when both x and y are approaching infinity it the limit can only reach positive OR negative infinity but not both, as is the case with y=x.

My question is, why can't we have a limit that goes from +x and -x to infinity and negative infinity, and likewise for the y values?

The only way that I could think of requires 3 assumptions.

1). We have a sphere with an infinitely large radius,

2). We can take the circumference of the sphere, and

3). If we were on the plane of the line of the circumference, it would be completely straight to us.

Okay, so we have the circumference of an infinite sphere, and it seems straight to us. This means that we can either go left or right, in the negative and positive directions respectively. Because of the infinite proportions of the line, as we move to the right x approaches infinity and as we go left x approaches negative infinity. Because the line is circular and connected, if we continue along the right side of the line we will eventually reach the spot which we called negative infinity, and vice versa for the negative as well.

Because x approaches infinity from the left and from the right, the limit exists: x-> ±infinity f(x) = infinity.

All that to say, one can take a limit that reaches infinity from both negative and positive directions with correct stipulations.

My question: Why can't we have a limit that goes from +x and -x to infinity, and likewise for the y values? Why can't we? I've always been taught that this is impossible, but I feel that this method is feasible. I ask you, the Khan Academy community, does this work, and what can we do with it?(4 votes)- I'm not sure how you are including 𝑦-coordinates into this because a function need not diverge as 𝑥 approaches ±∞. But the idea that in the real number line (𝑥-axis) ∞ "wraps around" to -∞ is not nonsensical! It is actually really creative that you have thought of that and in fact you're not the first one to wonder about this!

https://en.wikipedia.org/wiki/Real_projective_line

The real projective line incorporates this interpretation. To truly understand how this all works you would need a good background in*projective geometry*. I suggest you look into it!(5 votes)

- How can we differentiate between limits for functions that are infinite vs those that DNE?(4 votes)
- Actually, if you take 1/|x-2|, the limit is infinity, therefore the limit does NOT exist. Think of lim = infinity as a special case of the limit not existing. Consider this intentionally absurd statement (from W. Michael Kelley's Humongous Book of Calculus Problems): "the limit is that it's infinitely unlimited". Yeah, makes no sense. If the limit is infinity, it means there is no limit, because the value just keeps increasing without limit.(3 votes)

- If f(x) = 1/x, isn't it undefined when it approaches 0, how does it have a limit of infinity?(2 votes)
- No, it's undefined at 0
*because*its limit is infinity when it approaches 0 (from the right).(4 votes)

- I have a doubt that whether left limit of function ([x^2]-1)/x^2-1 as x tends to 1 is infinite or limit does not exist or both the same?( where [.] represent greatest integer function.)(3 votes)
- a limit tending to infinity and limit doesnot exist both are same or not. can u please explain it with an example(3 votes)
- Why can't I just go to calculus here? I already took precal. Can I just skip it?(1 vote)
- This particular topic was covered in both pre calculus and University calculus for me in college. I think its more of the applications that come next, you have to make sure you have a very strong base in working with limits of in differential and integral calculus you will be lost. Limits are the crux of all the material. If you can't manipulate and understand Limits, you won't be able to conceptually understand the basics of a derivative. It is important stuff, I would invest in the extra practice.(4 votes)

## 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.