Rule de l'Hôpital - Continuous and Differentiable

  • Thread starter mk9898
  • Start date
  • #1
109
9

Homework Statement


Prove that

##h: [0,\infty) \rightarrow \mathbb R, x \mapsto
\begin{cases}
x^x, \ \ x>0\\
1, \ \ x = 0\\
\end{cases}
##
is continuous but not differentiable at x = 0.

The Attempt at a Solution


To show continuity, the limit as x approaches 0 from the right must equal to 1. Meaning:

##\lim_{x \ \downarrow \ 0} h(x) = 1##.

##\lim_{x \ \downarrow 0} h(x) = \lim_{x \ \downarrow 0} x^x = 0^0 = 1## (I'm not too content with this part. Just plugging in 0 seems wrong to me and I would have to maybe use the definition of continuity?)

Differentiable:
##h'(0) = 0##
##\lim_{x \ \downarrow 0} \frac{f(x) - f(0)}{x-0} = \lim_{x \ \downarrow 0} \frac{x^x - 0^0}{x-0} = \lim_{x \ \downarrow 0} \frac{x^x - 1}{x} = \lim_{x \ \downarrow 0} x^{x-1}-1## does not exists due to 1/0. Therefore h(x) is not differentiable.
 
Last edited:

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
17,316
7,164
There is no left side. The function is not defined for ##x < 0##.
 
  • #3
109
9
Crap. Right. Fixed it.
 
  • #4
109
9
Can anyone give me some insight please?
 
  • #5
109
9
I made an error in my calculations. I think I got it now.
 
  • #6
Dick
Science Advisor
Homework Helper
26,263
619
##\lim_{x \ \downarrow 0} h(x) = \lim_{x \ \downarrow 0} x^x = 0^0 = 1## (I'm not too content with this part. Just plugging in 0 seems wrong to me and I would have to maybe use the definition of continuity?)

Differentiable:
##h'(0) = 0##
##\lim_{x \ \downarrow 0} \frac{f(x) - f(0)}{x-0} = \lim_{x \ \downarrow 0} \frac{x^x - 0^0}{x-0} = \lim_{x \ \downarrow 0} \frac{x^x - 1}{x} = \lim_{x \ \downarrow 0} x^{x-1}-1## does not exists due to 1/0. Therefore h(x) is not differentiable.

I'm not too happy with what you are doing there either. Maybe you should use that ##x^x=e^{x \ln x}## to get some believable arguments.
 
  • #7
109
9
Thanks for the response. I figured that my last step there is false. After L'Hospital I found the answer.
 

Related Threads on Rule de l'Hôpital - Continuous and Differentiable

  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
2K
Replies
2
Views
9K
Replies
6
Views
3K
  • Last Post
Replies
0
Views
966
  • Last Post
Replies
1
Views
2K
  • Last Post
2
Replies
31
Views
1K
Replies
3
Views
4K
Top