A few questions related to $a^x=x^a$

Question

I have a few related questions I would like some help with. The numbers involved can be assumed to be positive real numbers strictly greater than 1.

Given a number $a>1$:

1. What is the easiest way to show that: $$\lim_{x\to\infty} \frac{x^a}{a^x}=0$$
2. What is the easiest way to show that, for any particular $a$, there exists a number $x_0$ such that if $x>x_0$ then $a^x > x^a$
3. Is it possible to determine the smallest such $x_0$? I think I can show that for $a > e$ then $x_0 = a$ should work, though I'm not sure it's rigorous. And I don't really know what do to for $1<a<e$
4. Given two real numbers $1<a<b$, is it possible to show the existence of and find a minimal $x_0$ such that if $x>x_0$ then $a^x > x^b$ ?

In some of the above questions it may be easier or interesting to assume $x \in \mathbb{Z^+}$ instead of being real.

What I started with was showing that you need this condition: $$\frac{x}{\ln{x}} > \frac{a}{\ln{a}}$$

Since I think I can show that $\frac{x}{\ln{x}}$ is increasing for $a>e$, then I think this directly implies that $x_0 = a$ as discussed above. But this doesn't help for $a<e$ (which we can see easily with the example of $a=2$). And I haven't made much progress on question 4 at all.

Does the Lambert W function come into play here somehow?

Answer 1

1. Show that $\lim_{x\to \infty} (a\ln x - x\ln a) = -\infty$
2. Both $a^x$ and $x^a$ are continuous. At $x = 0$ which is higher? At $x = \infty$ which is higher? But at $x = a$, they cross.
3. Yes. There are finitely many solutions to $a^x = x^a$. $x_0$ is the highest. That uniquely defines the value $x_0$. There are many, many root-finding techniques that will allow you to identify the intersection to whatever precision you need.
4. Only small adjustments need to be made to (2) and (3) to answer this.

And finally, as Chrystomath has said, yes, you can use Lambert's function. But then, using Lambert's function to find the values is not any easier or faster or more accurate than finding it by root-finding techniques. Slapping a name on a function only makes it easier to locate what other people have discovered about it.