How to evaluate \lim_{x\rightarrow\infty} \root(n)(x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0})-x

Saige Shannon 2022-04-20 Answered
How to evaluate
limxxn+an1xn1++a0nx
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Answers (1)

Salvador Ayala
Answered 2022-04-21 Author has 10 answers

Alternatively, rewrite this limit as
limxxn+an1xn1++a0nx
Consider the Taylor expansion around 0 of 1+zn. We have
1+zn=1+1nz+O(z2)
Setting z=an1x++a0xn we see that z=O(1x), and hence
1+zz=1+1n(an1x++a0xn)+O(1x2)=1+an1n1x+O(1x2)
Thus we have
x(1+an1x++a0xnn1)=an1n+O(1x)
and we conclude
limxx(1+an1x++a0xnn1)=an1n.

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