I don't understand why this is true. I would like to see the process of converting this limit to e, to better understand this topic.
$\lim_{n \to \infty }\left({n^{2}+n \over n^{2}+1}\right)^{n} = e^1 = e$
I don't understand why this is true. I would like to see the process of converting this limit to e, to better understand this topic.
$\lim_{n \to \infty }\left({n^{2}+n \over n^{2}+1}\right)^{n} = e^1 = e$
Another way to show that $\lim_{n \to \infty }(1+\frac{1}{n^{2}})^n=1 $ is to use this "contra-Bernoulli-inequality" (as I call it):
If $n$ is a positive integer and $ 0 \le x < 1/n$ then $(1+x)^n \le 1/(1-x n)$.
This is readily proved by induction, writing it in the form $(1-x n)(1+x)^n \le 1$.
Setting $x = 1/n^2$, we get $(1+1/n^2)^n \le 1/(1-1/n) = 1+\frac{1}{n-1}. $
Let $y=\lim_{x\to\infty}(1+\frac{1}{x})^x$ then:
$\begin{align*}\ln y &= \ln \space[\lim_{x \to \infty} (1+\frac{1}{x})^x] \\ &= \lim_{x \to \infty}[\ln (1+\frac{1}{x})^x] \\ &= \lim_{x \to \infty}[x\ln (1+\frac{1}{x})] \\ &= \lim_{x \to \infty}[ \frac{\ln (1+\frac{1}{x})}{\frac{1}{x}}]\end{align*}$
L'Hopital's rule is now used because other wise the limit would result in $\frac{0}{0}$:
$ \begin{align*} \ln y &= \lim_{x \to \infty}[\frac{(\frac{\frac{-1}{x^2}}{1+\frac{1}{x}})}{\frac{-1}{x^2}}] \\ &= \lim_{x \to \infty} 1+\frac{1}{x} \\ &= 1\end{align*}$
We now have $\ln y = 1$, this implies $y=e$. This concludes the proof.
Are you familiar with $\lim_{n\to\infty}\left(1+{1\over n}\right)^n=e$ If so, do you see how to compare the limit you want with the one you know?
Note that $({n^{2}+n \over n^{2}+1})^{n}=({1+\frac{1}{n} \over 1+\frac{1}{n^{2}}})^n=\frac{(1+\frac{1}{n})^n}{(1+\frac{1}{n^{2}})^n}.$ Note also that $\lim_{n \to \infty }(1+\frac{1}{n})^n=e.$ On the other hand, $\lim_{n \to \infty }(1+\frac{1}{n^{2}})^n=\lim_{n \to \infty }e^{n\ln(1+\frac{1}{n^{2}})} =e^{\lim_{n \to \infty }\frac{\ln(1+\frac{1}{n^{2}})}{\frac{1}{n}}}.$ By L'Hosiptal Rule, $\lim_{n \to \infty }\frac{\ln(1+\frac{1}{n^{2}})}{\frac{1}{n}}=\lim_{n \to \infty }\frac{(\frac{1}{1+\frac{1}{n^{2}}})(-\frac{2}{n^3})}{(-\frac{1}{n^2})}=0.$ Combining the above the equalities, we have $\lim_{n \to \infty }(1+\frac{1}{n^{2}})^n=e^0=1.$ Therefore, $\lim_{n \to \infty }({n^{2}+n \over n^{2}+1})^{n} = \frac{\lim_{n \to \infty }(1+\frac{1}{n})^n}{\lim_{n \to \infty }(1+\frac{1}{n^{2}})^n}=\frac{e}{1}=e.$