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## Main Question or Discussion Point

When is the following equivalence valid?

$$\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))$$

I was told that continuity of f is key here, but I'm not positive.

This question comes up, for instance in one proof showing the equivalence of the limit definition of the number e to the definition of the inverse of the natural logarithm.

$$\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))$$

I was told that continuity of f is key here, but I'm not positive.

This question comes up, for instance in one proof showing the equivalence of the limit definition of the number e to the definition of the inverse of the natural logarithm.