Question 76
Let $f : [0, \infty) \to [0, \infty)$ be a continuous bijective function, such that
$$
\lim_{x \to \infty} \frac{f^{-1}(f(x)/x)}{x} = 1.
$$
a) Show that $\lim_{x \to \infty} \frac{f(x)}{x} = \infty$ and $\lim_{x \to \infty} \frac{f^{-1}(ax)}{f^{-1}(x)} = 1$, for any $a \gt 0$.
b) Give an example of a function $f$ that satisfies the conditions from the statement.
$$
\lim_{x \to \infty} \frac{f^{-1}(f(x)/x)}{x} = 1.
$$
a) Show that $\lim_{x \to \infty} \frac{f(x)}{x} = \infty$ and $\lim_{x \to \infty} \frac{f^{-1}(ax)}{f^{-1}(x)} = 1$, for any $a \gt 0$.
b) Give an example of a function $f$ that satisfies the conditions from the statement.