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AP Calculus Question: Limit Chain rule proof / Function-interchange of limits?
hello
Okay, so since this is my first question ever, I had no idea how this worked. For all the math geeks and/or erudite individuals out there, I would like to discover how this theorem is proven.
The Theorem is used for solving questions such as this one: As x approaches infinity, what is the limit of (1+ax)^1/x
It is stated as: If the limit of f(u) as u approaches b is L, and the limit of g(x) as x approaches a is b
Then, the limit of f(g(x)) as x approaches a is L.
1 Answer
- 3 years ago
I understand the theorem up until the limit of g(x) as x approaches a being plugged into the equation of the limit of f(u) as u approaches b.
How and why is lim ( u > b ) f(u) [Which is basically lim (u > lim (x > a)(g(x))) (f(u)) ] equal to lim (x > a ) f(g(x))?
In other words, how is f(lim (x > a)(g(x))) possible [The Function-interchange rule of limits] and when is it applicable?