Comment on Activity 3

For a positive value of ${\displaystyle x}$ we know that ${\displaystyle n}$th tail ${\displaystyle \to 0}$ and th tail. It follows that ${\displaystyle x^n/n!\to 0}$ as well. For a negative value of ${\displaystyle x}$, we may look instead at ${\displaystyle |x^n\text{/}n!|={\left|x\right|}^n\text{/}n!}$ and apply the result of part (a) — because ${\displaystyle \text{|}x\text{|}}$ is positive. If ${\displaystyle |x^n\text{/}n!|\to 0}$, then ${\displaystyle x^n/n!}$ also must approach ${\displaystyle 0}$ as ${\displaystyle n\to \infty }$.

This may seem a bit surprising if we look only at relatively small values of ${\displaystyle n}$. For example, for ${\displaystyle n=5}$, ${\displaystyle {10}^n=100\text{,}000}$ and ${\displaystyle {100}^n=10\text{,}000\text{,}000\text{,}000}$ — whereas ${\displaystyle n!}$ is only 120. But no matter how fast the exponentials ${\displaystyle x^n}$ start growing, the factorials always catch up and overtake the exponentials, eventually growing so much faster that their ratio decreases to ${\displaystyle 0}$. We already knew that exponentials (constant base, growing exponent) grow very fast. Now we see that factorials grow much faster. The result of Activity 3 is demonstrated in an entirely different way in Exercise 10.