Question

Sequences question

Answer 1

The sequence {\[a _{n}\]}, where \[a _{n}\] = \[\left( 1 + x/n \right)^{2}\] converges for all real numbers x to some number \[a _{x}\] > 0. Find \[a _{x}\] (your answer will be in terms of x)

Answer 2

It's not displaying correctly so here's a link: https://i.imgsafe.org/804ff67665.png

Answer 3

Do you remember the limit definition of the number e?\[\large\rm \lim_{n\to\infty}\left(1+\frac1n\right)^{n}=e\]Seems like that might be helpful here...

Answer 4

Am I looking in the right direction? at all?

Answer 5

\[\ln (x) = \int\limits_{1}^{x} \frac{ 1 }{ t } dt\]

Answer 6

and because \[\ln (e) = 1\], \[1 = \int\limits_{e}^{1} \frac{ 1 }{ t } dt\]

Answer 7

switch e and 1 in the equation above

Answer 8

\[\large\rm \lim_{n\to\infty}\left(1+\frac1n\right)^{n}=e\tag{1}\]

Answer 9

If we're allowed to use the above limit, its easy :
\[\large\rm \lim_{n\to\infty}\left(1+\frac x n\right)^{n}\]
letting \(n=xt\) gives
\[\large\rm \lim_{t\to\infty}\left(1+\frac{x}{ xt}\right)^{xt}\]