If you have a polynomial in the form: $$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0$$ where n is a positive integer and $a_n$ are constant real numbers. How come the end behavior is solely based on the degree of the polynomial (the n in:$x^n$) and the a in:$a_n$? In other words, why does the end behavior only depend on the degree, and the “a” multiplied by the x with the highest exponent? Why doesn’t the end behavior have anything to do with the following values of $a_{n-1}x^{n-1}...$?
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asked Oct 21, 2019 at 17:13
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1$\begingroup$ Because when $x$ gets very large, those other terms are simply negligible compared to the $a_nx^n$ term. $\endgroup$– 79037662Commented Oct 21, 2019 at 17:17
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2$\begingroup$ The short and dirty explanation is because for large enough values of $n$, $x^n$ is much larger than all of the other terms combined. For a more formal explanation, you can consider comparing the function $g(x)=a_nx^n$ to the function $f(x)$. Notice that $\lim\limits_{x\to\infty}\frac{f(x)}{g(x)}=1$, similarly for the other direction. $\endgroup$– JMoravitzCommented Oct 21, 2019 at 17:18
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1 Answer
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Consider the polynomial $x^2+x+1$ and evaluate it at increasing values:
$$10\to 111$$ $$100\to 10101$$ $$1000\to 1001001$$ $$10000\to 100010001$$ $$100000\to 10000100001$$ $$1000000\to 1000001000001$$ $$\cdots$$
What do you think about the "end behavior" ?