Limit

[Chiss]

How does one solve the following problem? I have tried calculating the limit of the function but am currently stuck. Symobal.com's calculator used L'hospital's rule but I could not get it to work. Help is appreciated.

Show that f is derivable in all x∈R and write the formula to f ′ (x)

 

[Zermelo]

Hi, you need to show us your work, and exactly where you’re stuck.

 

[Chiss]

If you have the following expression:



What would the application of L'hospital's rule look like?
The limit is


and when I try to apply L'hospital's rule I get something like this:

Do I use the wrong approach or have I implemented L'hospital's rule incorrectly?

 

[Steven G]

If you know L'hopital's rule then you know how to find derivative without using the definition for derivatives!

If y = e^u, then dy/dx = u' * e^u. So what is dy/dx in your case?

 

[Zermelo]

If you have the following expression:

View attachment 29048

What would the application of L'hospital's rule look like?
The limit is

View attachment 29049
and when I try to apply L'hospital's rule I get something like this:
View attachment 29051
Do I use the wrong approach or have I implemented L'hospital's rule incorrectly?

There is a theorem that states something along the lines of:
If f is a continious and derivable on (a, b), then the right derivative f’+(a) = lim (x-> a+) f’(x) (the right limit), so if you want to know if f has a derivative atm0, you need to show that f’-(0) = f’+(0) aka the left and right limits of f’ are exual, aka lim x->0 f’(x) exists