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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...

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  • $\begingroup$ Because $x^n \to \infty$ for $|x| > 1$ and 0 for $|x| < 1$, as $n \to \infty$ $\endgroup$
    – JuliusL33t
    Jun 6, 2018 at 8:19
  • $\begingroup$ The reason why it is near zero for $x\in(-1,1)$ is that when you exponentiate within that radius , the larger the exponent the smaller the values get . if $-1\lt x\lt 1$ then for larger powers the values decrease. if $x = -1$ then the values alternate if $x= 1$ then it stays $1$ $\endgroup$ Jun 6, 2018 at 8:19
  • $\begingroup$ You probably mean for $f_n\in C^0,f_n\to f$, $f\notin C^0$. You can also consider $f_n(x)=\frac1{1+x^n}$. $\endgroup$ Jun 6, 2018 at 8:21
  • $\begingroup$ @LittleCuteKemono I saw the graph it looks like a box $\endgroup$ Jun 6, 2018 at 8:22
  • $\begingroup$ @Mathstextbook Please recall that if the OP is solved you can evaluate to accept an answer among the given, more details here meta.stackexchange.com/questions/5234/… $\endgroup$
    – user
    Aug 3, 2018 at 21:56

2 Answers 2

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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 $n$.

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  • $\begingroup$ Exactly correct (except we should really look at the magnitude, $|x|<1$). Just for some extra information for the OP, consider the boundary case where $x = 1$. No matter how many times you multiply $1$ with itself it remains $1$, i.e. $1\times 1\times 1 \times ...\times 1 = 1$. Therefore $x = 1$ is just the point that separates the two different behavioral patterns of the function. $\endgroup$
    – Eff
    Jun 6, 2018 at 8:22
  • $\begingroup$ Absolutly right, I will add a positivity-constraint to my answer. $\endgroup$
    – Laray
    Jun 6, 2018 at 13:41
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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 $|a^n|\to \infty$
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