TITLE: Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
QUESTION [65 upvotes]: This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates.

What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$
In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} +\cdots +a_1 x+ a_0$, how would I find $$\lim_{x\rightarrow\infty} P(x)^{1/n}-x.$$
For example, how would I evaluate limits such as $$\lim_{x\rightarrow\infty} \sqrt{x^2 +x+1}-x$$ or $$\lim_{x\rightarrow\infty} \sqrt[5]{x^5 +x^3 +99x+101}-x.$$

REPLY [0 votes]: Let $n$ be an integer $\geqslant 2,$ and $q(y)$ a polynomial of
degree $\leqslant n - 2.$ Then $y^n + q(y) > 0$ for all sufficiently
large $y.$ For such $y,$ define $z = \sqrt[n]{y^n + q(y)}.$ Then
$z > 0,$ and
$$
y^{n-1}|z - y| \leqslant
|z^{n-1} + z^{n-2}y + \cdots + zy^{n-2} + y^{n-1}||z - y| =
|z^n - y^n| = |q(y)|,
$$
whence $z - y \to 0$ as $y \to \infty.$
Writing $x = y - a_{n-1}/{n},$ we have $y \to \infty$ as $x \to \infty,$ and so
$$
\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}} - x =
\sqrt[n]{y^n + q(y)} - y + \frac{a_{n-1}}{n} \to \frac{a_{n-1}}{n}.
$$
{"set_name": "stack_exchange", "score": 65, "question_id": 30040}