Theorem: Limit of a composition functionIf F and G are functions such that lim g(x) = L and lim f(x) = f(L), then….
lim f(g(x)) = f(lim g(x)) = f(L)
If F and G are functions such that lim g(x) = L and lim f(x) = f(L), then the limit of the composite function f(g(x)) as x approaches a is f(L), provided that f is continuous at L.
In other words, the limit of the composition of two functions is equal to the limit of the outer function evaluated at the limit of the inner function, provided that the outer function is continuous at that point.
This theorem is particularly useful in calculus and analysis, where it allows us to evaluate more complex limits by breaking them down into simpler parts. It also allows us to prove the continuity of composite functions, which is essential in many areas of mathematics and science.
As an example, let’s consider the following problem:
Find the limit of the function f(x) = sin(x^2) as x approaches 0.
We can think of f(x) as the composition of two functions: f(x) = sin(g(x)), where g(x) = x^2. We know from basic calculus that lim g(x) = 0 as x approaches 0. We also know that sin(x) is a continuous function at x=0. Therefore, we can apply the theorem to conclude that:
lim f(x) = lim sin(g(x)) = sin(lim g(x)) = sin(0) = 0
So the limit of f(x) as x approaches 0 is 0.
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