Behaviour of $x^n$ for very large $n$?

Question

when I plotted the graphs of $y=x^{10},x^{100},x^{1000}$ etc. I noticed that the shape approached an open rectangle with base between $x=-1$ and $x=1$,
But why does $x^n$ approaches this shape and is almost zero for $x \in (-1,1)$ and increases suddenly afterwards.
Are there any other functions which change behaviour suddenly, Please explain...

Answer 1

The fact is that

• for $|a|<1$ that is $-1<a<1$ we have $a^n\to 0$

whereas

• for $|a|>1$ that is $a<-1$ and $a>1$ we have $|a^n|\to \infty$

Answer 2

The function $x^n$ is the repeated multiplication of $x$ with itself. Let's focus on positive values $x>0$ first, the negatives are a little hassle but not too hard, if you understood the concept.

As long as $x<1$ holds, this value will get smaller every time you mutliply it. Take $0.9$ for example $$ 0.9>0.81>0.729>0.6561>... $$

The contrary is done, when $x>1$ holds true. In that case, you enlarge the value with each additional multiplication. Take $1.1$: $$ 1.1<1.21<1.331<1.4641<... $$ This difference will increase, as you increase $n$.