For a positive value of `x` we know that `n`th tail `rarr 0` and `0<x^n//n!<n`th tail. It follows that `x^n//n! rarr 0` as well. For a negative value of `x`, we may look instead at `|x^ntext[/]n!|=|x|^ntext[/]n!` and apply the result of part (a) — because `text[|] x text[|]` is positive. If `|x^ntext[/]n!| rarr 0`, then `x^n//n!` also must approach `0` as `n rarr oo`.
This may seem a bit surprising if we look only at relatively small values of `n`. For example, for `n=5`, `10^n=100text[,]000` and `100^n=10text[,]000text[,]000text[,]000` — whereas `n!` is only 120. But no matter how fast the exponentials `x^n` start growing, the factorials always catch up and overtake the exponentials, eventually growing so much faster that their ratio decreases to `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.