Asymptotic notation question

Question

Arrange the following functions in increasing asymptotic order:

Answer 1

In addition to JMoravitz's comment, note that:

$$\lim \limits_{n \to infinity} \frac{1.0000001^n}{n^a}$$ is infinite. So we can apply L'Hopital's Rule. The derivative of $$1.0000001^n = 1.0000001^n ln(1.0000001)$$ and the derivative of $$n^a = an^{a-1}$$ which is still infinity in the limit that $n$ goes to infinity. The $a^{th} $ iteration of L'Hopital's rule would yield: $$\lim \limits_{n \to infinity} \frac{1.0000001^n ln(1.0000001)^a}{a!}$$ which is still infinity. Therefore, $1.0000001^n$ is asymptotically larger than any finite degree polynomial, despite its appearance.