# Comparing composite functions asymptotically

Suppose, we have $f_1(n) = o\left(f_2(n)\right)$. (small $o$)

Now, we have to compare $f_1\left(f_2(n)\right)$ and $f_2\left(f_1(n)\right)$ asymptotically..

What can we say about it? Which one will be greater & why?

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## 2 Answers

HINT $f_1 = \log x,\; f_2 = x^n \;$ for $\; n = 1/2,\; 1,\; 2\;$ shows that the answer is "nothing"

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What happens if $f(x)=x^t$ and $g(x)=e^x$ where $t$ is a positive real?

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