# Thread: help with limits of sequences

1. ## help with limits of sequences

Hey everyone

I am a bit confuse about the following questions especially about the first and the second one

I need to find each sequence's limit and it's pretty urgent

please I really need help with those three

thanks

2. ## Re: help with limits of sequences

$$\left(1-\frac1{n^2}\right)^n = \left(1-\frac1{n^2}\right)^{(n^2\frac{n}{n^2})} = \left(\left(1-\frac1{n^2}\right)^{n^2}\right)^{\frac1n}$$
The expression inside the (outer) brackets is easily transformed into a standard result.

Similar manipulations deal with the other examples. For the second, write $$n^2+1=(n^2-1)+2$$

3. ## Re: help with limits of sequences

Originally Posted by someone111888
the first and the second one

I need to find each sequence's limit and it's pretty urgen
I suggest for the second one you might consider:
${\left( {\dfrac{{{n^2} + 1}}{{{n^2} - 1}}} \right)^{{n^2} + 3n - 16}} = {\left( {1 + \dfrac{2}{{{n^2} - 1}}} \right)^{{n^2}}}{\left( {1 + \dfrac{2}{{{n^2} - 1}}} \right)^{3n - 16}}$

4. ## Re: help with limits of sequences

but how can I solve it from here?

5. ## Re: help with limits of sequences

Originally Posted by someone111888
but how can I solve it from here?
Suppose $\displaystyle S = \lim_{n\to \infty}s_n$.

Then $\displaystyle \ln S = \ln \lim_{n\to \infty} s_n = \lim_{n\to \infty} \ln (s_n)$