• Support PF! Buy your school textbooks, materials and every day products Here!

Limit with ln in the exponent

  • Thread starter Kachiko
  • Start date
  • #1
4
0

Homework Statement


lim as x approaches 0 of (x + 1)ln x

Homework Equations





The Attempt at a Solution


I tried using ln y = (ln x)[ln (x + 1)]
then:
ln (x + 1)
----------
1
---
ln x
to make it eligible for L'Hopital's Rule. Then differentiating the numerator and the denominator, I got:
1
---
x + 1
-------
-1
---------
x(ln x)2
Then I brought the denominator of the denominator up:
-x(ln x)2
----------
x + 1

The answer is supposed to be 1. Therefore my differentiating should have evaluated to 0. Doesn't my answer give -infinity? Please help. I suck at ln and e. I must have missed something.
 
Last edited:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
-x(ln x)^2/(x+1) is still an indeterminant form as x->0, ln(x)->-infinity. You'll have to apply l'Hopital again to figure out the limit of the numerator.
 
  • #3
4
0
Thank you for replying.

I did that, and I got: - 2 ln x - (ln x)2, which is inf - inf. I'm rather confused how to convert that to numerator/denominator form again, though.
 
Last edited:
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
You want the limit of -x*ln(x)^2. First write it as -ln(x)^2/(1/x). I'm not sure how you got what you got.
 
  • #5
4
0
Ah. My mistake. You said "numerator". I hadn't realized that once you'd found the limit of either numerator or denominator you could go to work on each separately (I thought if you used L'Hopital you always had to differentiate both haha).

Anyway, after a couple more differentiations of the numerator I eventually got -2x which would indeed make the whole thing zero. Thanks. :!!)
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
618
Right. Once you know the limit of a part of an expression you can set it aside and just work with the rest using l'Hopital. It simplifies the expressions.
 

Related Threads for: Limit with ln in the exponent

Replies
4
Views
4K
  • Last Post
Replies
4
Views
474
Replies
2
Views
1K
  • Last Post
Replies
17
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
5K
Replies
2
Views
13K
Top