Find the limit of the following sequences or determine that the limit does not exist. {Bn} where Bn = {n/(n+1) if n 5000

Question

Find the limit of the following sequences or determine that the limit does not exist. {Bn} where Bn = {n/(n+1) if n < or equal to 5000 {ne^-n if n>5000

anonymous · Answer

Since it is a sequence, we're taking the limit as $n\to\infty$.

anonymous · Answer

yes ok

anonymous · Answer

We're going to use that $ne^{-n}$ case since that is the case used as we approach $\infty$.

anonymous · Answer

alright

anonymous · Answer

Looking at it straight up, we get  $\infty e^{-\infty}\to \infty\times 0$ which is an indeterminate form.

anonymous · Answer

Well $$
\frac{n}{e^n}
$$Can be solved with l'Hospital's rule.

anonymous · Answer

idk the d/dx of e^n

anonymous · Answer

Suppose $$
f(n)=a_n$$where $f(n)$ is a continuous function.
Then I'm pretty sure: $$
\lim_{n\to \infty }a_n = \lim_{x\to \infty} f(x)
$$when the limit exists.

anonymous · Answer

In fact, the definition of a limit and sequence are the same so yeah, they are the same.

anonymous · Answer

So treat it like a continuous function and differentiate with respect to $n$ .

anonymous · Answer

whats the final answer?

anonymous · Answer

What is the derivative?

anonymous · Answer

1/e^x.

anonymous · Answer

Okay so what does that limit approach?

anonymous · Answer

5000