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Limits of composite functions: internal limit doesn't exist

Finding the limit of g(h(x)) at x=-1 when the limit of h(x) at x=-1 doesn't exist. Does it mean that the composite limit doesn't exist? Not necessarily! See how we analyze it. Created by Sal Khan.

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• Shouldn't it be -3 from the left, so -3^- ?
(39 votes)
• It also took me some time to understand. Basically, he is shortening the stuff.

When you take the limit of h(x) from the right-hand side (x=>1^+) you get -2.

When you take the limit of h(x) from the right-hand side (x=>1^-) you get -3.

After that when you do the same for g(h(x)) from the left and right-hand side for h(x=>-2), you will get 3 for both hand sides.

Same for the h(x)=>-3, both left and right-hand sides will be the same, with the answer of 3.

instead of doing the whole stuff he just did the desired and required stuff.
(12 votes)
• It is confusing from while taking the LH limit of g(h(x)) as x app-1^- and equating to LH limit of g(h(x)) as h(x) app -3 #should it be -3^- or -3^+. Please answer.
(11 votes)
• It also took me a while to figure out:

When you're thinking about what h(x) is approaching, you are looking at the y (vertical) axis .
It'd be as if you tilt the whole image by 90º (to the right), or tilt your head to the left like an owl. And so, on the y axis:
"right" (+) is the upper side of the graph,
"left" (-) is the lower side of the graph.

So if you look at this graph, h(x) is approaching both -2 and -3 from the right / upper (+) side.
You don't have to look at both + and - for each. You only have to figure out where it's approaching from.

Hope that helps!
(23 votes)
• After an initial confusion, I think I have understood the procedure, but I would like confirmation that I have done it right, because I am not sure.
Towards the end, Sal calculates the limit of h(x) as x approaches -1 from the left. That is easy: -3.
But then the complicated thing comes, because to find the corresponding limit of g(x), looks for the limit from the right (!). What I think is the following:
We have -3 we got from the previous step. But if we look at the graph, the values of y are diminishing as they approach that point. That is, -3 is approximated from above.
Now, when we plug this value in the x-axis of g(x), we need to do it in the same way, that is, from above, i.e., from the values greater than -3, which means that we are approximating from the right.
Can anybody confirm that I have understood it right? Thanks.
(10 votes)
• Correct. I find it healthier to think of from the right as "values above x " i.e. x⁺ and from the left as "values below x " i.e. x⁻.
(2 votes)
• Stuck at h(x) -> -3+, checked the comments here and on YT, people say its + and not - because we're looking at the Y values. I thinks that's right but how do we know which value to look at? In previous videos I was under the impression that we considered the X axis when determining right hand or left hand?
(8 votes)
• I think considering the Y-axis is just for when we are doing composite functions because in a composite function, we are taking the outputs/y-values of the internal function and then using them as inputs/x-values of the external function.

For other operations such as addition, subtraction, multiplication and division, the y-values remained as y-values, they never became x-values for another function which is why it was a lot more straight forward.

Here are steps that I found from someone else (I can't seem to find their comment anymore) that really helped me understand what was going on:

1) Take the limit of the function h(x) as x -> -1 from the right

2) The OUTPUT of that function will be -1.9, -1.99, -1.999...

3) That sequence of numbers will serve as the INPUT for g(x): g(-1.9), g(-1.99), g(-1.999)...

4) Following that same series of inputs in g(x) as x -> -2 means you get closer to -2 from the RIGHT/POSITIVE side of the graph

5) Do the same thing for the left side and. we see that x seems to be approaching -3 from both sides

Hope this helps and If I made a mistake anywhere please be sure to correct me!
(4 votes)
• I'm in 6th grade, so this is taking a bit to comprehend.
(5 votes)
• `h(x)` has no limit as it approaches the x-position of -1. From the left side, the function seems to be approaching y = -3, while from the left side, the function seems to be approaching y = -2. Since the "left and right limits" don't match, the function doesn't have one single limit.

However, we are not focusing on the limit of `h(x)` by itself, we are focusing on the composite function `f(h(x))`. The limit of `h(x)` from the left side is -3, so plugging that into our function `f(h(x))` would result in `f(-3)` from the left side, since `h(x)` = 3 (as the function approaches -1 from the left).

The limit of `h(x)` from the right side is -2, so that would be `f(-2)`. In the graph `f(x)`, the limit of f as it approaches -2 and the limit of `f(x)` as it approaches -3 are both equal to 3, so the limit of the composite function `f(h(x))` is equal to three.

I hope you understand! If I made anything unclear, please let me know. Also, nice job taking the initiative and learning Calculus in 6th grade (I am learning it early too)!
(4 votes)
• Why do you approach -3 from the right?
(6 votes)
• When we say
``lim g(h(x)) when h(x) --> -2^+``

it's the same as saying
``lim g(y) when y --> -2^+``

We are now dealing with the y (vertical) axis instead of the x (horizontal) axis, because `h(x)` is the y. So, the positive (right) of the y axis is from above and the negative (left) of the y axis is from below, that's why it's
``lim g(h(x)) when h(x) --> -2^+&lim g(h(x)) when h(x) --> -3^+``
and not
``lim g(h(x)) when h(x) --> -2^+&lim g(h(x)) when h(x) --> -3^-``
because they are both approaching from above (positive side of the y (vertical) axis).

Conclusion: When dealing with the y axis of a limit, flip the graph 90 degrees to the right.

Additionally: The right side is the side of the positive numbers, and the left side is the side of the negative numbers.

Hope this helps
(4 votes)
• This is confusing. Inputting something that does not exist into a function should yield something that does not exist. How come it actually exists?
(3 votes)
• The meaning of a limit that DNE just means that the one-sided limits are different i.e. value of the function as x approaches a from below is different from the value of the function as x approach a from values above a. But this is only true for the inner function h(x).

We know that the lim x→-1 g(h(x)) exists and is true so long if lim x→-1⁺ g(h(x)) = lim x→-1⁻ g(h(x)). We just need to prove that the one-sided limits for the composite function are the same for the limit of the composite function to exist.
The composite function is taking the output of the inner function as input.
As x→-1⁺ for h(x), the output h(x)→-2⁺. Plugging this h(x) output as the x-input into g(x), lim h(x)→-2⁺ g(h(x)) = 3

As x→-1⁻ for h(x), h(x)→-3⁺. Plugging this h(x) output as the x-input into g(x), lim h(x)→-3⁺ g(h(x)) = 3

To visualise this, I found it very helpful to visualise this on the graphs. Look at the video again, notice Sal uses the arrows in both h(x) and g(x).
As x→-1⁺, draw an arrow of h(x)→-2⁺. Now look at g(x) which takes h(x) as its x-input.
For g(x), as x (now x is h(x)) → -2⁺, g(x) = 3.
Now go back to h(x). As x→-1⁻, limit h(x) → -3⁻ and draw an arrow as h(x) → -3⁻.
Now go to g(x) which takes h(x) as its x-input, as x (now is equal to h(x)) → -3⁺, g(x) = 3

So we have proven that lim x→-1 g(h(x)) exists and is equivalent to: lim x→-1⁺ g(h(x)) = lim x→-1⁻ g(h(x)) because the one-sided limits are the same.

Avoid being overly fixated on the thought that the limit that DNE. DNE just means the one-sided limits are different. Just because the one-sided limits are different for one function does not mean that the one-sided limits of another function (which could take these different one-sided limits as input) are different.
Thus, even if a limit does not exist (one-sided limits are different) for one function, does not mean that the limit for another function who takes these different one-sided limits as input, does not exist.
(3 votes)
• When solving for the composite limit, he made both of the limit tests right handed limit tests when it came to testing the limit of g(x). Is that because they have to be the same direction in order to see if they're truly the same value? Or is that some other thing I should be thinking about? (If my question doesn't make sense, I'm asking why he used right bound limit tests for both the left and right bound limit tests after he found the value of h(x) under the the limit x->-1)
(2 votes)
• Hey Tyler, I think I had the same question as you. If you're confused about why he did the lim h(x) -> -2+ then the lim h(x) -> -3+ (and thought they were both right handed limits) the confusion happens because lim h(x) -> some value is not the same as lim x -> some value. To my understanding when you have a limit h(x) -> some value, it's the limit when the Y-value of h(x) approaches some value, not the x value.

Basically the function g(h(x)) is taking the y-values of h(x) and plugging them into g(x). Let's take the example lim x -> -1+ g(h(x)):

Firstly, you plug in x values getting infinitely close to -1 from the right into h(x). So that would be -0.9, -0.99, -0.999 etc. You see that the values will approach -2 from above, hence the notation that h(x) -> -2+.

I think you are able to change the original lim x-> -1+ g(h(x)) to lim h(x) -> -2+ g(h(x)) for that same reason. You are taking the x value, putting it into h(x), and the outputs (y-values) approach -2+(from above) are being inputted into g(x). Therefore, you take those outputted y-values and sub them into g(x) (like you would for any composite function). It would look something like this

g(-1.9) g(-1.99) g(-1.9999) getting infinitely close to g(-2) from values greater than -2 as possible, because that's how the outputs for h(x) are behaving when you do h(-0.9) h(-0.99) etc. keep in mind the original lim is x-> -1+ g(h(x)). All of that gives the final answer of 3.

The ^+ on the -2 and -3 denote the behavior (above and below) of the y-values of h(x).

I'm not sure if that's exactly correct but that's how I've understood it so far.
(5 votes)
• What happens when you get two different answers for both the g(h(x))?
(3 votes)
• Then the limit does not exist
(2 votes)

Video transcript

- [Instructor] All right, let's get a little bit more practice taking limits of composite functions. Here, we want to figure out what is the limit as x approaches negative one of g of h of x? The function g, we see it defined graphically here on the left, and the function h, we see it defined graphically here on the right. Pause this video and have a go at this. All right, now your first temptation might be to say, all right, what is the limit as x approaches negative one of h of x, and if that limit exists, then input that into g. If you take the limit as x approaches negative one of h of x, you see that you have a different limit as you approach from the right than when you approach from the left. So your temptation might be to give up at this point, but what we'll do in this video is to realize that this composite limit actually exists even though the limit as x approaches negative one of h of x does not exist. How do we figure this out? Well, what we could do is take right-handed and left-handed limits. Let's first figure out what is the limit as x approaches negative one from the right hand side of g of h of x? Well, to think about that, what is the limit of h as x approaches negative one from the right hand side? As we approach negative one from the right hand side, it looks like h is approaching negative two. Another way to think about it is this is going to be equal to the limit as h of x approaches negative two, and what direction is it approaching negative two from? Well, it's approaching negative two from values larger than negative two. H of x is decreasing down to negative two as x approaches negative one from the right. So it's approaching from values larger than negative two of g of h of x. G of h of x. I'm color coding it to be able to keep track of things. This is analogous to saying what is the limit, if you think about it as x approaches negative two from the positive direction of g? Here, h is just the input into g. So the input into g is approaching negative two from above, from the right I should say, from values larger than negative two, and we can see that g is approaching three. So this right over here is going to be equal to three. Now, let's take the limit as x approaches negative one from the left of g of h of x. What we could do is first think about what is h approaching as x approaches negative one from the left? As x approaches negative one from the left, it looks like h is approaching negative three. We could say this is the limit as h of x is approaching negative three, and it is approaching negative three from values greater than negative three. H of x is approaching negative three from above, or we could say from values greater than negative three, and then of g of h of x. Another way to think about it, what is the limit as the input to g approaches negative three from the right? As we approach negative three from the right, g is right here at three, so this is going to be equal to three again. So notice the right hand limit and the left hand limit in this case are both equal to three. So when the right hand and the left hand limit is equal to the same thing, we know that the limit is equal to that thing. This is a pretty cool example, because the limit of, you could say the internal function right over here of h of x, did not exist, but the limit of the composite function still exists.