volume of a cube and a ball in n-dimensions?

In very high dimensional space, the opposite corners of a cube are very far from one another. The unit cube is a huge object as compared to the unit sphere, where every point is 2 units or less from every other point.

A first remark: For a cube of side $a$ in $n$ dimensions, the volume is $a^n$. Thus the limit depends on whether $a\in(0,1)$, $a=1$ or $a>1$.

You are comparing the volumes $B_n$ of the ball of radius one and $C_n$ of the cube of side two. The reason why $C_n/B_n\to\infty$ as $n\to\infty$ is that, loosely speaking, much of the volume of the cube is near its corners which the ball doesn't reach, and this phenomenon becomes stronger and stronger as dimension grows. In other words, a ball of radius one cannot go very far in every direction at once, but the cube of side two can.