Answer
$x^{10}$ grows slower than $e^{0.01x}$.
Work Step by Step
We will find the limit $\lim_{x\to\infty}\frac{x^{10}}{e^{0.01x}}.$
1) If it is equal to zero then $x^{10}$ grows slower than $e^{0.01x}$;
2) If it is equal to $\infty$ then $x^{10}$ grows faster than $e^{0.01x}$;
3) If it is equal to some constant the their growth rates are comparable.
$$L=\lim_{x\to\infty}\frac{x^{10}}{e^{0.01x}}=\lim_{x\to\infty}\left(\frac{x}{e^{\frac{0.01x}{10}}}\right)^{10}=\left(\lim_{x\to\infty}\frac{x}{e^{0.001x}}\right)^{10}=l^{10},$$
where we denoted $l=\lim_{x\to\infty}\frac{x}{e^{0.001x}}.$ Now we have
$$\lim_{x\to\infty}\frac{x}{e^{0.001x}}=\left[\frac{\infty}{e^\infty}\right]=\left[\frac{\infty}{\infty}\right][\text{LR}]=\lim_{x\to\infty}\frac{(x)'}{(e^{0.001x})'}=\lim_{x\to\infty}\frac{1}{0.001e^{0.001x}}=\left[\frac{1}{0.001e^\infty}\right]=\left[\frac{1}{\infty}\right]=0.$$
Now returning to the initial limit we have
$$L=l^{10}=0^{10}=0,$$
and thus $x^{10}$ grows slower than $e^{0.01x}$.
