Question

L'Hopital's Rule Problem:
lim(x->0) (1/((x^(1/5))(lnx)))

I know the answer is -infinity, but I can't figure out how to get there.

Answer 1

\[\lim_{x\to 0}\frac{1}{\sqrt[5]{x}\ln(x)}\]?

Answer 2

Yes, sorry if it was unclear..

Answer 3

and it is approaching 0^(+)

Answer 4

ok well you cannot use l'hopital as it is , since it is not in the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) we will have to rewrite it some how to make it one of those

Answer 5

usual gimmick is to take the reciprocal of one of those things
i would pick rewriting as
\[\lim_{x\to 0}\frac{1}{\sqrt[5]{x}\ln(x)}=\lim_{x\to 0}\frac{\frac{1}{\sqrt[5]{x}}}{{\ln(x)}}\]

Answer 6

I wrote it as (1/x^(1/5))/ln(x)... basically what you did..

Answer 7

now it looks like \(\frac{-\infty}{-\infty}\) and you can use l'hopital

Answer 8

oh ok then you should be in good shape

Answer 9

so that's what I got, but apparently the answer should be -infinity.. not positive infinity.

Answer 10

hmm

Answer 11

ok i'm going to write out what I worked out.. and please tell me if it's correct.. i'll use that equation drawing thing

Answer 12

how come i am getting zero? am i doing something wrong?

Answer 13

oh yeah i am

Answer 14

ok this drawing is taking way too long.. I got:
(d/dx)(1/x^(1/5)) = -1/(5*(x^(6/5)) then d/dx of lnx is 1/x

Answer 15

working that out I got 0 at first, then I made some random adjustments to the order (putting ln(x)) in the numerator and ended up with positive infinty

Answer 16

yeah your exponent is wrong, it should be negative

Answer 17

it is.. i wrote -1/... making it negative

Answer 18

\[\large \frac{1}{x^{\frac{1}{5}}}=x^{-\frac{1}{5}}\] the derivative is
\[\large -\frac{1}{5}x^{-\frac{6}{5}}\]

Answer 19

you get
\[\frac{-\frac{1}{5}x^{-\frac{6}{5}}}{\frac{1}{x}}\]

Answer 20

clean it up and you get
\[-\frac{1}{5x^{\frac{1}{5}}}\]

Answer 21

so if you have -(1/0) that gives you -infin?

Answer 22

yup

Answer 23

denominator is going to zero through positive numbers, since you are taking
\[\lim_{x\to 0^+}\]

Answer 24

and you have a minus sign out front, making the whole thing negative

Answer 25

so i just spent the past 30 minutes redoing this problem because i didn't know that.. i feel.. inadequate.. thank you for the help, though!!

Answer 26

lol
yw