Limits problem

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leslielau

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Sep 7, 2010
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Hi Everyone, I'm new to the forums. I'm currently stuck on the following problem about limits :

Express the following limits in terms of e

1) lim (1+1/(n+1) )^-n
n->infinity

For this one I don't know where to start...

2) lim (1+[3/n]+[2/(n^2)])^n
n->infinity

For this one, I got to the part where I simplified the fractions by writing it as lim [(n+1)(n+2)/(n^2)]^n, and then I'm stuck.
n->inf

Are there any special ways to deal with these limits ? Thanks for all the help !!
 
For the second one,note it has the same limit as \(\displaystyle \lim_{n\to \infty}\left(1+\frac{3}{n}\right)^{n}\)
 
I don't understand why they are equal ... can you explain it ? Thanks a lot !!
 
leslielau said:
Hi Everyone, I'm new to the forums. I'm currently stuck on the following problem about limits :

Express the following limits in terms of e

1) lim (1+1/(n+1) )^-n
n->infinity

For this one I don't know where to start...

2) lim \left (1+ 3/n + 2/n^2)^n
n->infinity

For this one, I got to the part where I simplified the fractions by writing it as
lim [(n+1)(n+2)/(n^2)]^n, and then I'm stuck.
n->inf

Are there any special ways to deal with these limits ? Thanks for all the help !!
Click to expand...

Do you know how to write e[sup:20lkubmt]x[/sup:20lkubmt] as a limit of a function (similar to the first problem)

\(\displaystyle \lim_{n \to \infty} \left (1+\frac{3}{n} + \frac{2}{n^2}\right )^n\)

\(\displaystyle = \lim_{n \to \infty} \left (\frac{n+1}{n} * \frac{n+2}{n}\right )^n\)

\(\displaystyle = \lim_{n \to \infty} \left (\frac{n+1}{n}\right )^n * \lim_{n \to \infty} \left (\frac{n+2}{n}\right )^n\)

\(\displaystyle = \lim_{n \to \infty} \left (1 + \frac{1}{n}\right )^n * \lim_{n \to \infty} \left (1+ \frac{2}{n}\right )^n\)

Now it is same problem as (1)....
 
Ok Thanks ! I tried (2) and got the answer e^3 and then I tried (1) and got 1/e. Are they correct ?
 
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