## anonymous 3 years ago Tricky limit?

1. anonymous

$\lim_{x\to 0}\left(\frac{(1+x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}}.$

2. anonymous

sorry, thats kind of hard to see.

3. anonymous

$\LARGE \lim_{x\to 0}\left(\frac{(1+x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}}.$

4. anonymous

$\Large e = \lim_{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^n$When you reparameterize: $$x=1/n$$ $\Large e = \lim_{x \rightarrow 0}\left(1+x\right)^\frac{1}{x}$

5. anonymous

So it's a matter of settling that outer 1/x

6. anonymous

I haven't learned 'reparameterize' yet, so I would assume I wouldn't have to use that... Supposed to use L'Hospital's rule

7. anonymous

Okay, then you need to have an indeterminate form of $$\infty /\infty$$ or $$0/0$$

8. anonymous

right.

9. anonymous

|dw:1353045336700:dw|

10. anonymous

|dw:1353045499277:dw|

11. anonymous

@mahmit2012 I am supposed to use L'Hospital's rule.

12. anonymous

I can't read that bottom line.

13. anonymous

"and in this case it is 1/sqrt(e)

14. anonymous

No, of the previous picture

15. anonymous

I don't know

16. anonymous

Okay, let's start with bringing in the $$1/x$$ to the numerator and denominator and figuring out if that is an indeterminate form.

17. anonymous

|dw:1353047217870:dw|

18. anonymous

|dw:1353047297672:dw|

19. anonymous

@waterineyes can you help me understand this maybe?