## Yahoo! 2 years ago $\lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x }{ n })^{\frac{ 1 }{ x }}$

1. Yahoo!

Power is For Both nume and Deno

2. hartnn

can be solved by taking log on both sides...

3. hartnn

and also without LH, do you know what lim x->0 [a^x-1 /x] =... ?

4. hartnn

*[a^x-1]/x

5. Yahoo!

ln a/b

6. hartnn

$\lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x }{ n })^{\frac{ 1 }{ x }}\\=\lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x +n-n}{ n })^{\frac{ 1 }{ x }} \\ \lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x+n }{ n }-1)^{\frac{ 1 }{ x }} \\\lim_{x \rightarrow 0}(\frac{ 1^x-1 + 2^x -1+ 3 ^x-1 ........n^x-1 }{ n }-1)^{\frac{ 1 }{ x }}$ got the hint ?

7. hartnn

and you need to take log on both sides first....to get 'x' in the denominator.

8. hartnn

am i on right track? @experimentX

9. experimentX

seems so ... ;let me do on copy!!

10. experimentX

oh .. that last one is +1

11. hartnn

yep, typo...

12. hartnn

$\lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x }{ n })^{\frac{ 1 }{ x }}\\=\lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x +n-n}{ n })^{\frac{ 1 }{ x }} \\ \lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x-n }{ n }+1)^{\frac{ 1 }{ x }} \\\lim_{x \rightarrow 0}(\frac{ 1^x-1 + 2^x -1+ 3 ^x-1 ........n^x-1 }{ n }+1)^{\frac{ 1 }{ x }}$

13. experimentX

although ... L'hopital seems a bit faster!!

14. hartnn

LH after taking log, right ?

15. experimentX

yeah!! your method works just fine as well!!

16. hartnn

hmm...much faster! thanks~

17. experimentX

18. experimentX

should be n! e^(1/n)

19. experimentX

woops!! sorry (n!)^(1/n)

20. hartnn

using the standard formula for lim x->0 (1+x)^(1/x) and adjusting the exponents .

21. experimentX

the other would be use the usual unusual trick!! x = e^ln(x)

22. hartnn

haven't practised that much...would like to see steps, if you don't mind...

23. hartnn

why do i feel i have asked this limit Q..... O.o, @Yahoo! where are you ?

24. Yahoo!

I am Here..:)

25. Yahoo!

i also..Need to know the steps Plzz

26. experimentX

|dw:1357499465779:dw|

27. experimentX

|dw:1357499559730:dw|

28. experimentX

|dw:1357499607470:dw|

29. experimentX

that's same as @hartnn 's |dw:1357499736048:dw|

30. experimentX

x = e^ln(x) is the usual trick for types of f(x)^g(x) like ... n->0 (n!)^(1/n^n)

31. hartnn

got your steps :) thank you!

32. experimentX

well :)

33. satellite73

i can't think of another way at all you are going to have to either take the log or rewrite as an exponential (which is really the same thing) because you have a variable in the exponent

34. Yahoo!

|dw:1357578098421:dw|

35. Yahoo!

Then ??

36. experimentX

|dw:1357578176065:dw|

37. Yahoo!

Got it..Thxx