Question

indeterminate forms

Answer 1

okay

Answer 2

so since \(a\) can be any number, so can these limits

Answer 3

this is a big one :/ I am honestly stumped!

Answer 4

oh you have to prove that these limits are all \(a\) ?

Answer 5

well it just says to show that they can have any positive real value :|

Answer 6

but i don't understand the problem!

Answer 7

\[\huge x^{\frac{\ln(a)}{1+\ln(x)}}\]
\[=\huge e^{\frac{\ln(a)}{1+\ln(x)}\ln(x)}\] is a start for all of them

Answer 8

oh ok yes! i actually had that written! :D

Answer 9

but thats as far as i got

Answer 10

use l'hopital to find the limit in the exponent
it is \(\ln(a)\) and so the limit is \(e^{\ln(a)}=a\)

Answer 11

the thing you need to show here is that
\[\lim_{x\to 0}\frac{\ln(a)\ln(x)}{1+\ln(x)}=\ln(a)\]

Answer 12

don't forget of course that \(\ln(a)\)is just a constant, so this is the same as showing
\[\lim_{x\to 0}\frac{\ln(x)}{1+\ln(x)}=1\]

Answer 13

…hmm okay I'm trying to process this :|

Answer 14

you can use l'hopital right?

Answer 15

yes! so do i perform that on the limit of ln (x) / (1+ ln(x)) ?

Answer 16

yes, and you can to it in your head since it is more or less obvious that the derivative of both is \(\frac{1}{x}\) making the ration \(1\)

Answer 17

*ratio

Answer 18

yes!

Answer 19

so basically is all that the question is asking me to prove?

Answer 20

seems to simple! lol :s

Answer 21

so with the \(\ln(a)\) you get the limit of \(\ln(a)\) making the entire limit \(e^{\ln(a)}=a\)

Answer 22

yeah, kinda

Answer 23

hard part is figuring out what you need to do
the mechanics is not much

Answer 24

hm ok. so for part B i just follow the same steps?