proving a polynomial limit: If P(x) has degree r>0, show....

[shakalandro]

I need to prove this problem...

If P(x) is a polynomial of degree r>0, show that the limit of P(n+1)/P(n) = 1.

 

[PAULK]

Re: proving a polynomial limit

You omitted a tiny detail :

lim(as x --> what?)

probably you meant x --> infinity?

 

[shakalandro]

Re: proving a polynomial limit

Yes, as n approaches infinity.

 

[galactus]

Re: proving a polynomial limit

If we have a poly of degree r, then when taking the limit as n-> inf we can disregard the smaller powers.

\(\displaystyle \lim_{n\to \infty}\frac{(n+1)^{r}}{n^{r}}=\lim_{n\to \infty}\left[\frac{r}{n}+\frac{C(r,2)}{n^{2}}+\frac{C(r,3)}{n^{3}}+.....+\frac{C(r,r)}{n^{r}}+1\right]=\lim_{n\to\infty}\sum_{k=0}^{r}\frac{C(r,k)}{n^{k}}\)

As \(\displaystyle n\to\infty\), everything goes to 0 and we are left with 1.

 

[BigGlenntheHeavy]

Common sense should dictate that the limit is one as n/n = 1, n a real number.

Note: Galactus, I realize this is not a rigourous proof, just musing.