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This is one of those "dumb" questions. When solving a problem recently I found that the key to solving it was to interchange the limit operator and the exponential operator. Because the function happened to be continuous, this could be achieved.

But I am still in the dark as to why continuity allows for interchanging of those two operators. I don't see the direct causal link.

In addition, what other operators would continuity let us interchange? And which would not be amenable to continuity?

(The problem can be found here: Prove that $\lim_{n\to \infty} \left(1+a_n(x/n)\right)^n=1$ given that $\lim_{n\to \infty} a_n = 0$..)

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David
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1 Answers1

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Continuity is defined as $$ \lim_{x \to a} f(x) = f(a). $$ An equivalent formulation is that for any Cauchy sequence $(x_n)$, $x_n \to a$, we have $$ \lim_{n \to \infty} f(x_n) = f(a) = f\left( \lim_{n \to \infty} x_n \right), $$ which says precisely that we can swap the limit and the sum.

(To prove the equivalence, if you have one Cauchy sequence that doesn't work, you get counterexample when you try and find the $\delta$ so that $|f(x)-f(a)|<\delta$. For the other direction, if $f$ does not tend to the limit, you have a Cauchy sequence of $\varepsilon$s and a set of corresponding $x$ values that violate the same inequality. So basically you show "$f$ does not tend to a limit" $\iff$ "there is a Cauchy sequence on which $f$ does not tend to a limit", and the contrapositive of this is the result I gave.)

Chappers
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