How would one compute the following limit?
$$\lim_{n \to \infty} \left( 1 – \frac{1}{n} \right)^{n}$$
I know
$$\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^{n} = e$$
but right there is a minus keeping that limit from being used.
Another problem I am questioning is finding the limit of
$$\frac{n!}{2n}$$
Of course, $\frac{x^n}{n!}$ has zero as a limit but here it is the opposite.
Best Answer
You can obtain the limit of
$$\left(1-\frac1n\right)^n$$
easily from the one you know, $\left(1 + \frac1n\right)^n \to e$, by noting
$$1 - \frac1n = \frac{n-1}{n} = \frac{1}{\left(\frac{n}{n-1}\right)} = \frac{1}{\left(1 + \frac{1}{n-1}\right)}.$$
Then you can write
$$\left(1 - \frac1n\right)^n = \left(1-\frac1n\right)\frac{1}{\left(1+\frac{1}{n-1}\right)^{n-1}},$$
where the first factor obviously converges to $1$, and the second one converges to $\frac{1}{e}$ by what you already know.