Limits on Composite Functions- Appears DNE but has a limit

In summary, the conversation discusses the use of limit laws in finding the limit of a composition of functions, specifically f(x) + g(x) as x approaches -2. The expert summarizer explains that the limits of the component functions (f(x) and g(x)) are irrelevant and only the limit of the combined function (h(x)) is important. The expert also clarifies that the limit laws require a limit to exist for what is being started with and for each "piece" that the expression is being broken into. Finally, the expert explains that in this case, the limit of h(x) exists as the two individual limits from the left and right approach the same value.
  • #1
opus
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Please see my attached image, which is a screenshot from Khan Academy on the limits of composite functions.

I just want to check if I'm understanding this correctly, particularly for #1, which has work shown on the picture.

Now my question:
We are taking the limit of a composition of functions, namely f(x) + g(x).
Now for the limit laws to work, as I understand, a limit has to exist with what you're starting with, and a limit has to exist for what you turn it into. In other words, a limit must exist in the RHS and LHS.
Now for problem 1, the limit of f(x) as x approaches -2 clearly does not exist. And the limit of g(x) as x approaches -2 does not exist. So how can we use the limit laws on this? A limit doesn't exist at the value that we are taking the limit of for each of the functions in the composition?

In the video, he goes on to take the limit of f(x) and g(x) as x approaches -2 from the left, and separately from the right. And these limits each sum to 4, so then it is said that the limit of the composition is 4.
I do understand this reasoning, but what isn't sitting right with me is the fact that the limits didn't exist at x=-2 in the first place, but by going at x=-2 from each side individually, the limit now exists.
 

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  • #2
Edit: I was using the term "composite" but what I actually went was "Combined".
 
  • #3
opus said:
We are taking the limit of a composition of functions, namely f(x) + g(x).
Now for the limit laws to work, as I understand, a limit has to exist with what you're starting with,...
Which is a function ##h\, : \,\mathbb{R} \longrightarrow \mathbb{R}## obtained as ##h(x) := \left( f(x)+g(x) \right)## - brackets first!
... and a limit has to exist for what you turn it into. In other words, a limit must exist in the RHS and LHS.
What do you mean by LHS and RHS? The limit and the ##4\,##?
Now for problem 1, the limit of f(x) as x approaches -2 clearly does not exist. And the limit of g(x) as x approaches -2 does not exist. So how can we use the limit laws on this? A limit doesn't exist at the value that we are taking the limit of for each of the functions in the composition?

In the video, he goes on to take the limit of f(x) and g(x) as x approaches -2 from the left, and separately from the right. And these limits each sum to 4, so then it is said that the limit of the composition is 4.
I do understand this reasoning, but what isn't sitting right with me is the fact that the limits didn't exist at x=-2 in the first place, but by going at x=-2 from each side individually, the limit now exists.
The limits of the component functions are irrelevant. Only if they existed, then we would have ##\lim h= \lim f +\lim g##. Here we are talking about ##h(x)##. Strictly speaking we would have to distinguish
$$
\lim_{x \to -2 +0} h(x) = \lim_{x \to -2 +0}(f(-2)+g(-2))=3+1
$$
and
$$
\lim_{x \to -2 -0} h(x) = \lim_{x \to -2 -0}(f(-2)+g(-2))=1+3
$$
In any case, we have to consider the function ##h(x)## which adds up differently from the right and from the left, but as the sums are equal, the limit exists.

##h(x)## is not continuous at ##x=-2##. If we approach its value from the left or from the right, we get the limit ##4##. But we have ##h(-2)=6##, so ##h(x)## around ##x=-2## is a line where the function value at ##x=-2## is two units above the rest.
 
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  • #4
fresh_42 said:
What do you mean by LHS and RHS? The limit and the 44\,?
What is mean is the Right Hand Side of the Equation and the Left Hand Side of the equation. In other words, if I have the limit of an expression on the right hand side, and want to use the limit laws to break it up, a) a limit must exist for what we are starting with (on the LHS), and a limit must exist for each "piece" we break it up into (on the right hand side).

fresh_42 said:
The limits of the component functions are irrelevant. Only if they existed, then we would have limh=limf+limg\lim h= \lim f +\lim g. Here we are talking about h(x)h(x).
Ok so since the limit of ##f(x)## as x approaches ##-2## doesn't exist, and the limit of ##g(x)## as x approaches ##-2## doesn't exist, we cannot say that the ##lim~h = lim~f + lim~g## because they don't individually have limits, so there are no limits to add?

By ##h(x)##, you are referring to the function that is the combination of ##f(x)+g(x)##?
And let me spell this out to see if I understand correctly:

We have ##\lim_{x \rightarrow -2} {f(x) + g(x)}##
##\lim_{x \rightarrow -2} {f(x)}## Does Not Exist
##\lim_{x \rightarrow -2} {g(x)}## Does Not Exist
However, this is irrelevant because we want the limit of h(x) which is their combination.
So what we can do is take ##\lim_{x \rightarrow -2^-} {f(x)} + \lim_{x \rightarrow -2^-} {g(x)} = L##
Then we can take ##\lim_{x \rightarrow -2^+} {f(x)} + \lim_{x \rightarrow -2^+} {g(x)} = M##
If ##L=M##, that means that as ##h(x)##, which is the combination of ##f## and ##g##, approaches ##-2## from the left and right, then ##\lim_{x \rightarrow -2}{h(x)}## EXISTS?
 
  • #5
opus said:
What is mean is the Right Hand Side of the Equation and the Left Hand Side of the equation. In other words, if I have the limit of an expression on the right hand side, and want to use the limit laws to break it up, a) a limit must exist for what we are starting with (on the LHS), and a limit must exist for each "piece" we break it up into (on the right hand side).
Yes, but without mention the equation you talk about, this is pretty senseless. All I saw was ##\lim_{x \to -2} (f(x)+g(x))=4## and it didn't appear as if you would talk about ##4##. There is simply no RHS to talk about!
Ok so since the limit of ##f(x)## as x approaches ##-2## doesn't exist, and the limit of ##g(x)## as x approaches ##-2## doesn't exist, we cannot say that the ##lim~h = lim~f + lim~g## because they don't individually have limits, so there are no limits to add?
Firstly, we have to distinguish right from left approaches here. Secondly, you are right. One of the two limits ##\lim f## and ##\lim g## doesn't exist on neither side. The formula is not applicable.
By ##h(x)##, you are referring to the function that is the combination of ##f(x)+g(x)##?
No. By ##x \mapsto h(x)## I was referring to exactly what I defined. I only used ##f(x)## and ##g(x)## for every single value of ##x##. I did not use the functions as such, neither did I compose anything. I picked values ##f(a)## and ##g(a)##, i.e. real numbers, added them - as numbers - and named the result ##h(a)##. As I did this for all values ##a##, I got a function ##a \mapsto h(a)##, only that I named my new variable ##x## instead of ##a##. But I did not use any property of ##f## or ##g## which says they are functions.
And let me spell this out to see if I understand correctly:
As long as you refuse to distinguish between left and right, any answer to this is risky, because it allows misunderstandings. What do you mean by "do not exist"? In our case both limits exist (left and right), but they are not the same, so the limit as such does not exist, correct, but this is for the technical requirement, that we only speak of a limit (if left and right aren't mentioned), if they coincide.
We have ##\lim_{x \rightarrow -2} {f(x) + g(x)}##
We have ##\lim_{x \to -2} \left[\dfrac{}{} \quad f(x)+g(x) \quad\right]##
##\lim_{x \rightarrow -2} {f(x)}## Does Not Exist
##\lim_{x \rightarrow -2} {g(x)}## Does Not Exist
Yes, because left and right do not match. Otherwise they do exist.
However, this is irrelevant because we want the limit of h(x) which is their combination.
We want the limit of the expression in square brackets. Combination is unfortunate here, as it suggests we would need the two components. We only use them to define a sum, that's all.
So what we can do is take ##\lim_{x \rightarrow -2^-} {f(x)} + \lim_{x \rightarrow -2^-} {g(x)} = L##
Then we can take ##\lim_{x \rightarrow -2^+} {f(x)} + \lim_{x \rightarrow -2^+} {g(x)} = M##
If ##L=M##, that means that as ##h(x)##, which is the combination of ##f## and ##g##, approaches ##-2## from the left and right, then ##\lim_{x \rightarrow -2}{h(x)}## EXISTS?
Yes. Since a) the existence of those individual limits is given, so that the formula for addition is applicable in each individual case, and b) the technical requirement of being equal in order to speak of a limit is fulfilled.

Short:
##\lim_{x \to -2^+}f(x) \text{ defined }\; , \; \lim_{x \to -2^-}f(x) \text{ defined }\; , \;\lim_{x \to -2}f(x) \text{ undefined }##
##\lim_{x \to -2^+}g(x) \text{ defined }\; , \; \lim_{x \to -2^-}g(x) \text{ defined }\; , \;\lim_{x \to -2}g(x) \text{ undefined }##
##\lim_{x \to -2^+}h(x) \text{ defined }\; , \; \lim_{x \to -2^-}h(x) \text{ defined }\; , \;\lim_{x \to -2}h(x) \text{ defined }##
##\lim_{x \to -2} h(x) = 4 \neq 6 = h(-2)\; , \;h(x) \text{ is discontinuous }##
 
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  • #6
To summarize what @fresh_42 wrote in his short version:
##\lim_{x \to -2^-}f(x) = 1## and ##\lim_{x \to -2^+}f(x) = 3##
Therefore, ##\lim_{x \to -2}f(x)## does not exist.
Similarly, since the left- and right-hand limits for g(x) as x approaches -2 are different, ##\lim_{x \to -2}g(x)## also does not exist, so the two-sided limit of f(x) + g(x) doesn't exist as x approaches -2.
 
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  • #7
Here's a sketch, not exactly the actual values, i.e. I haven't checked the slope or so, but that's the crucial point here:
upload_2018-9-10_14-0-24.png
 

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  • #8
Mark44 said:
Similarly, since the left- and right-hand limits for g(x) as x approaches -2 are different, limx→−2g(x)limx→−2g(x)\lim_{x \to -2}g(x) also does not exist, so the two-sided limit of f(x) + g(x) doesn't exist as x approaches -2.
It seems that I was looking only at the consitutent functions f and g. It's very possible that lim(f + g) exists at a particular point, even though lim f and lim g don't exist at that point.
 
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  • #9
Thanks for the responses guys. Still trying to dissect this a bit. Going to take a mental breather from this and come back to it tonight for a proper reply. :DD
 

Related to Limits on Composite Functions- Appears DNE but has a limit

What are composite functions and their limits?

A composite function is a combination of two or more functions where the output of one function becomes the input of another. The limit of a composite function is the value that the function approaches as the input approaches a certain value.

Can a composite function have a limit even if it appears to not exist?

Yes, a composite function can have a limit even if it appears to not exist. This can happen when the individual component functions have limits, but when combined, they do not exist due to the function becoming undefined at that point.

How do you determine if a composite function's limit exists or not?

To determine if a composite function's limit exists, you need to evaluate the limits of the individual component functions. If all the limits exist and are equal, then the composite function's limit exists at that point. However, if any of the individual limits do not exist or are not equal, then the composite function's limit does not exist.

What is the difference between a one-sided limit and a two-sided limit for a composite function?

A one-sided limit only considers the behavior of the function from one direction, either the left or the right. A two-sided limit takes into account the behavior of the function from both directions, the left and the right. For a composite function, both the one-sided and two-sided limits must exist and be equal for the overall limit to exist.

Can a composite function have a limit at one point but not at another?

Yes, a composite function can have a limit at one point but not at another. This can occur if the individual component functions have different limits at that point, causing the composite function's overall limit to not exist. It is important to evaluate the limits of each component function separately to determine if the composite function's limit exists at a certain point.

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