Question

lim x goes to infiniti, (x+1)^((1/(ln(x+1))

Answer 1

according to wolframalpha, the answer is e, but i cant figure out how

Answer 2

use l'hopital 's rule and it will eventually be e

Answer 3

\[\lim_{x\rightarrow\infty}(1+x)^{\frac{1}{\ln(x+1)}}\]

Answer 4

step one is as usual take the log

Answer 5

you get
\[\frac{1}{\ln(x+1)}\times \ln(x+1)\]

Answer 6

first take down the exponent, and put it like
1/(ln(x+1)) * (ln(x+1))

Answer 7

that limit is pretty obviously one, since it is identically one

Answer 8

but I got e on my CAS software

Answer 9

so your answer is
\[e^1=e\]

Answer 10

ooooh that is right :-D

Answer 11

wait the answer is e, not 1
step one was to take the log
step two is compute the limits
step three is exponentiate since step one was take the log

Answer 12

huh? im confused with step one? do you mind breaking it down into baby steps?

Answer 13

ok... foo! lol... I'm determined .. ha ha...

\[y = \ (x+1)^{1/\ln (x+1)}\]

\[\ln y = \ln (x+1)^{1/\ln (x+1)}\]

\[\ln y = (1/\ln (x+1))(\ln (x+1))\]

\[e ^{\ln y} = e ^{\ln (x+1)/\ln (x+1)}\]

y = e^1

dy/dx = e^1 = e

Answer 14

where did you get the y from?