What is the proof for nested limits?

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swampwiz
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.

Homework Helper
S.G. Janssens
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.

swampwiz
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.