If $p(x)$ is a polynomial then $\lim_{k \to \infty}\frac{p(k+1)}{p(k)}=1$

Question

Let $p(x)$ be a polynomial of degree $n$, i.e $p(x)=a_nx^n+a_{n-1}x^{n-1}+...a_0, a_n\neq0$

For every $\varepsilon >0$, we have to find a stage such that after that stage it should be like $${\left|\frac{p(k+1)}{p(k)}-1\right|<\varepsilon},$$

i.e. $${\left|\frac{p(k+1)-p(k)}{p(k)}\right|=\frac{|a_n((k+1)^n-k^n))+...a_2((k+1)^2-k^2)+a_1|}{|a_nk^n+a_{n-1}k^{n-1}+...a_0|}}$$

I dont know how to proceed further

Answer 1

$$\lim_{k\to\infty}\frac{p(k+1)}{p(k)} =\frac{a_n.(k+1)^n+a_{n-1}(k+1)^{n-1}....a_1}{a_n.k^n+a_{n-1}k^{n-1}....a_{n-1}}$$

$$=\frac{a_n.(1+{1 \over k})^n+{a_{n-1} \over k}(1+ {1 \over k})^{n-1}....{a_1 \over k^n}}{a_n+{a_{n-1} \over k}+....{a_1 \over k^n} }$$ $$=\frac{a_n}{a_n}=1$$

Answer 2

Note that for $x \neq 0$, ${p(x) \over x^n} = a_n + a_{n+1} {1 \over x} + \cdots$.

Hence $\lim_{x \to \infty} {p(x) \over x^n} = a_n$.

Hence $\lim_{x \to \infty} {p(x+1) \over p(x)} = \lim_{x \to \infty}{{p(x+1) \over (x+1)^n} \over {p(x) \over x^n} } = {a_n \over a_n} = 1$.