## DLS 2 years ago Limits question [Challenge]

1. DLS

$\LARGE \lim_{x \rightarrow 0} \frac{(1+x)^\frac{1}{x} -e}{x}$

2. DLS

@yrelhan4 @shubhamsrg

3. shubhamsrg

* :)

4. DLS

first time in life :O

5. ParthKohli

Taylor Series expansion?

6. DLS

Use whatever you want.

7. ParthKohli

Ans 0?

8. DLS

Nope

9. experimentX

Taylor series works

10. ParthKohli

Weird. The Taylor Series expansion is in the terms of $$e$$ and $$x$$. The first term is $$e$$

11. DLS

Taylor series gives 0?

12. experimentX

gives -e/2 if you calculate correctly

13. ParthKohli

Oh wait! lol, yeah. I forgot to consider that -e/2 which had no $$x$$

14. DLS

How about a different way rather than taylor series?

15. ParthKohli

Definition of derivative? :-|

16. DLS

o.O

17. ParthKohli

I was just thinking how this might be the definition of a derivative in disguise.

18. experimentX

Binomial expansion might work ... but that's too long

19. ParthKohli

DLS Use whatever you want. 15 minutes ago DLS How about a different way rather than taylor series? 9 minutes ago

20. DLS

You are right,still investigating for more methods,I wrote so to specify that you are not bound to any particular method,just asking if you know anything alternate :)

21. experimentX

L'hopital works

22. experimentX

http://www.wolframalpha.com/input/?i=Limit%5B%281+%2B+x%29%5E%281%2Fx%29+%281%2F%28x+%281+%2B+x%29%29+-+Log%5B1+%2B+x%5D%2Fx%5E2%29%2C+x+-%3E+0%5D after using L'hopital you get this expression. The first term tends to e .. if you again apply L'hopital on the second term, you will get -1/2