## amyna one year ago find the limit of the sequence n-1/n

1. amyna

you take the limit as n approaches infinity

2. amyna

but i don't know how to start

3. zepdrix

Is it $$\large\rm \frac{n-1}{n}$$ or $$\large\rm n-\frac{1}{n}$$ ?

4. amyna

the first one

5. zepdrix

Oh limit of the sequence? not the series? Ok that makes things a little easier then :d

6. zepdrix

$\large\rm \lim_{n\to\infty}\frac{n-1}{n}\quad=\quad \lim_{n\to\infty}\frac{n}{n}-\frac{1}{n}\quad=\quad \lim_{n\to\infty}1-\lim_{n\to\infty}\frac{1}{n}$You could break it up like this I suppose. That seems like a good start :o

7. amyna

ok i will try this approach, can i also multiply both sides by n ?

8. zepdrix

No no let's not try anything fancy like that :) We don't actually have any "sides" here. It's just this guy $$\large\rm \lim\limits_{n\to\infty}\frac{n-1}{n}$$ and we're rewriting him in a way that's easier to handle. Our limit depends on n, So in the same way that you can't factor n out of the limit, Example:$$\large\rm \lim\limits_{n\to\infty}2\frac{1}{n}\ne\frac{1}{n}\lim\limits_{n\to\infty}\frac{1}{n}2$$ (Can't pull the n stuff out like that) we also don't want to introduce new n stuff into the limit! :o

9. zepdrix

Ah typo in my example :c woops

10. zepdrix

I was trying to say that you can't factor the 1/n outside of the limit. Was just trying to give an example why you don't want to multiply any n's into the limit.

11. amyna

ok got it

12. zepdrix

$\large\rm =\quad \lim_{n\to\infty}1-\lim_{n\to\infty}\frac{1}{n}$But if we were able to break it down into these two limits, these should be manageable. You don't remember that second limit? :o

13. amyna

then how would i solve it after i break the function up? sorry it has been long since i learned limits so i don't really remember how to actually solve it.

14. zepdrix

I applied a limit rule to break it into two separate limits. As n approaches infinity, what does the number 1 approach?

15. amyna

infinity as well ?

16. zepdrix

careful! we're not adding up a bunch of 1's. We're just looking at that number 1 as n grows and grows. 1 is a constant. It's unchanging. So as n gets larger and larger, 1 approaches 1. Hopefully that makes some sense :o

17. zepdrix

But for the other part... As n approaches infinity... let's look at a really big n to see what's going on.$\large\rm \lim_{n\to\infty}\frac{1}{n}\approx\frac{1}{12bajillion}$I plugged in a really really big number for n, letting it get "close" to infinity.

18. zepdrix

This is one of those limits that you really want to remember :) But oh well hehe. So do you notice anything about this number? How about a number you can plug into your calculator, like 1/99999?

19. amyna

so how would i show the necessary work for this problem since it approaches 1?

20. zepdrix

You do these simple algebra steps,$\large\rm \lim_{n\to\infty}\frac{n-1}{n}\quad=\quad \lim_{n\to\infty}\frac{n}{n}-\frac{1}{n}\quad=\quad \lim_{n\to\infty}1-\frac{1}{n}$And then, at this point in your class, your teacher probably wants you to state what rule you're using here. $\large\rm =\lim_{n\to\infty}1-\lim_{n\to\infty}\frac{1}{n}$Justify that step by saying something like... The limit of a difference is equivalent to the difference of limits. Or whatever the name of that limit rule is, I can't remember. $\large\rm =1-0$$\large\rm =1$I think that's all you need to do :o I don't think you need to say anything special about the 1/n limit. That's a pretty common one. I could be wrong though.

21. amyna

Okay this makes so much more sense now! Thank You for your help!

22. zepdrix

yay team \c:/