Question

\[\lim_{x \rightarrow 0}(\frac{ 1^x + 2^x + 3 ^x ........n^x }{ n })^{\frac{ 1 }{ x }}\]

Answer 1

Power is For Both nume and Deno

Answer 2

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

Answer 3

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

Answer 4

*[a^x-1]/x

Answer 5

ln a/b

Answer 6

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

Answer 7

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

Answer 8

am i on right track? @experimentX

Answer 9

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

Answer 10

oh .. that last one is +1

Answer 11

yep, typo...

Answer 12

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

Answer 13

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

Answer 14

LH after taking log, right ?

Answer 15

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

Answer 16

hmm...much faster! thanks~

Answer 17

oh!! i had been using unnecessary approximations. your method is faster!!

Answer 18

should be n! e^(1/n)

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

I am Here..:)

Answer 25

i also..Need to know the steps Plzz