baldymcgee6 3 years ago Tricky limit?

1. baldymcgee6

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

2. baldymcgee6

sorry, thats kind of hard to see.

3. baldymcgee6

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

4. wio

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

5. wio

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

6. baldymcgee6

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. wio

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

8. baldymcgee6

right.

9. mahmit2012

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10. mahmit2012

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11. baldymcgee6

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

12. wio

I can't read that bottom line.

13. baldymcgee6

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

14. wio

No, of the previous picture

15. baldymcgee6

I don't know

16. wio

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. mahmit2012

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18. mahmit2012

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19. baldymcgee6

@waterineyes can you help me understand this maybe?