# Limit -- Variation on a problem

#### greg1313

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$$\displaystyle \displaystyle\lim_{n \to \infty}\bigg[n*e \ - \ n*(1 + \tfrac{1}{n})^n\bigg]$$

1 person

#### HallsofIvy

MHF Helper
First note that the discrete limit, $$\displaystyle \lim_{n\to \infty} ne- n\left(n+ \frac{1}{n}\right)^n$$ is the same as the continuous limit $$\displaystyle \lim_{x\to\infty} xe- x\left(x+ \frac{1}{x}\right)^x$$ as long as the latter limit exists. To do that, write it as $$\displaystyle \lim_{x\to\infty}\frac{2- \left(x+ \frac{1}{x}\right)^x}{\frac{1}{x}}$$, where both numerator go to 0, and use L'Hopital's rule.

#### Plato

MHF Helper
First note that the discrete limit, $$\displaystyle \lim_{n\to \infty} ne- n\left(\color{red}n+ \frac{1}{n}\right)^n$$ is the same as the continuous limit $$\displaystyle \large\lim_{x\to\infty} xe- x\left(\color{red}x+ \frac{1}{x}\right)^x$$ as long as the latter limit exists. To do that, write it as $$\displaystyle \large\lim_{x\to\infty}\frac{\color{red}2- \left(\color{red}x+ \frac{1}{x}\right)^x}{\frac{1}{x}}$$, where both numerator go to 0, and use L'Hopital's rule.
Prof. Ivey, are there typo in the above?

#### HallsofIvy

MHF Helper
Not actually a typo- I just misread the problem!

#### Idea

we can use L'Hôpital's rule a couple of times or write

$$\displaystyle e-\left(1+\frac{1}{n}\right)^n=\sum _{k=2}^n \left(\frac{1}{k!}-\left( \begin{array}{c} n \\ k \end{array} \right)\frac{1}{n^k}\right)+\sum _{k=n+1}^{\infty } \frac{1}{k!}$$

using the series for $e$ and the binomial theorem to expand $\left(1+\frac{1}{n}\right)^n$

do some algebra then multiply by $n$ and take the limit

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