l'hopital's rule (1 Viewer)

[CM_Tutor]

Originally posted by KeypadSDM
Lim[x->0] x^x
= Lim[x->0] e^xLn[x]

Lim[x->0] xLn[x]
= Lim[x->0] x * Lim[x->0] Ln[x]

D'oh.

Actually, you don't need the rest to do this limit - dominance rules dictate that polynomials dominate logs, so the behaviour of x will dictate the outcome.

Thus, lim (x->0) xln x = 0

You could also note that y = x^x has a minimum turning point at (1 / e, e^-1/e), meaning lim (x->O⁺) x^x > 0, and hence lim (x->O⁺) dy/dx = - inf. So, y = x^x is descending nearly vertically from the y-axis to a minimum, and then increases.

 

[Grey Council]

Originally posted by KeypadSDM
Screw it, I didn't use it

*forgets about this weird rule*