# Homework Help: Finding a limit

1. Dec 4, 2012

### chief10

1. The problem statement, all variables and given/known data

I'm having trouble with these here.. it's been a while since I've done sequences and I can't seem to make this work with Standard Limits equations.

Clearly the answer given by Wolfram solver is there after the = but i'd like to know the reasoning behind it.

Anyone that could point me in the right direction would be most helpful!

1.

2.

3.

thanks a lot

-chief10

Last edited: Dec 4, 2012
2. Dec 4, 2012

### haruspex

Do you know the standard limit result for (1+1/n)n? Yo can use that to solve 1 and 3.

3. Dec 4, 2012

### clamtrox

All you need to know is that $e^x = \lim_{n\rightarrow \infty} (1 + \frac{x}{n})^{n}$ and that if $\lim f(x) = a$ and $\lim g(x) = b$ with a,b≠∞, then $\lim f(x) g(x) = a b$

4. Dec 4, 2012

### chief10

so you just divide inside the brackets by 'n' right?

i don't get how the (n+1) part works with the particular standard limit though.. it's throwing me off a bit..

also what do you do with n^n..

5. Dec 4, 2012

### chief10

ah yeah just your standard limit laws

how do you apply it to this one though..

Last edited: Dec 4, 2012
6. Dec 4, 2012

### chief10

I couldn't work it out

SL 7 didn't work for me.. any more tips?

7. Dec 4, 2012

8. Dec 4, 2012

### dextercioby

You can also use the definition of e in terms of a limit for point 2, so you can have a common starting point for all 3. But using e for point 2 might be an overkill, though.

9. Dec 4, 2012

### happysauce

Actually for the third one you don't need the definition of e.

0≤n$^{n}$/(3+n)$^{n+1}$≤ ?? which approaches 0.

10. Dec 5, 2012

### chief10

This is what I've done for (1-(1/(n^2)))^n ===> I think it works

exp[lim(n→∞)n*log(1-(1/(n^2)))

let t = 1/n and use L'Hopitals

exp[2*lim(t→0)(t/((t^2)-1))

then take the limit of each individual component inside the exponent and you should be left with exp(2*0/0-1) = exp(0) = 1

reckon i should do the same thing [exp(log)] type approach for the others?

11. Dec 5, 2012

### chief10

could you elaborate further? i've never seen that concept before

12. Dec 5, 2012

### chief10

for number 1 i can't seem to solve it using exp[log]...

L'Hopitals doesn't work since the denominator is 1..

13. Dec 5, 2012

### clamtrox

Number 1 might be conceptually easier if you first write m = n+2 and consider m→∞. You can split it into two fractions, after which the rule I mentioned earlier becomes useful.

14. Dec 5, 2012

### happysauce

You know that the sequence is positive, if you show that the sequence is bounded by another sequence and that bounded sequence approaches 0, then you can deduce that your original sequence is 0. Similar concept to squeeze theorem.

15. Dec 5, 2012

### happysauce

For the second one you can do in 1 step. Factor it then apply the definition of e^x and you get 1/e * e = 1.