What is the proof for nested limits?

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  • #1
swampwiz
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AIUI, this is a law of proofs:

lim x→a f( g( x ) ) = f( lim x→a g( x ) )

I have searched for an explanation of this proof, but have been unable to find one, although I did find a page that was for certain types of functions of f( x ), just not a proof for a function in general.
 

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  • #2
lurflurf
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Not true in general. You need continuity assumptions. Even if all the limit exists, then still it is possible to have inequality, as playing with simple examples should indicate you.
 
  • #4
swampwiz
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of interest
https://teachingcalculus.com/2019/08/26/limit-of-composite-functions/

It is not true in general. That theorem is in any calculus book requiring lim x→a g( x ) exist and f continuous at that value.
In many examples of interest only one of the two conditions hold. We then need to find another method in those cases.

I was referring to functions that are the same expression, not some contrived function that is defined by different expressions for different sections of the domain.
 
  • #5
S.G. Janssens
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I was referring to functions that are the same expression, not some contrived function that is defined by different expressions for different sections of the domain.
That is a rather arbitrary criterium. Take any function ##F## on ##\mathbb{R}## that is, in your opinion, contrived. Take another function ##G## on ##\mathbb{R}## that you don't find contrived. Define the function ##H## on ##\mathbb{R}## by ##H(x) = F(G(x))##. Then ##H## can be written using one and the same expression for all ##x \in \mathbb{R}##. Do you find ##H## to be contrived, or not?
 
  • #6
zinq
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Suppose that g(x) is continuous near x = a and f(u) is continuous near u = g(a). Then by definition the limit of f(g(x)) as x → a is what happens to f(g(x)) when x gets closer and closer to a. This means we are plugging into g(x) values of x close to a. By continuity of g(x) near x = a, these g(x)'s will get closer and closer to g(a). So whatever g(a) may be, the limit of f(g(x)) as x → a is the same as the limit of f(u) as u → g(a). (Are you with me so far?)

Now since f(u) is continuous at u = g(a) the limit of f(u) as u → g(a) is the same as plugging in g(a) for u, or in other words f(g(a)).
 

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