lim-x-0-x-x-x-

Question Number 60006 by meme last updated on 17/May/19

l i \underset{x \to 0}{m} \frac{x^{x}}{x} = ?

Commented by maxmathsup by imad last updated on 17/May/19

l e t A (x) = \frac{x^{x}}{x} f o r x > 0 \Rightarrow A (x) = \frac{e^{x l n (x)}}{x} w e h a v e {l i m}_{x \to 0^{+}} e^{x l n (x)} = 1 b u t

e^{u} \sim 1 + u \Rightarrow e^{x l n x} \sim 1 + x l n (x) \Rightarrow \frac{e^{x l n (x)}}{x} \sim \frac{1}{x} + l n (x) = \frac{1 + x l n (x)}{x} \to + \infty (x \to 0^{+})

\Rightarrow {l i m}_{x \to 0^{+}} A (x) = + \infty

a n o t h e r w a y w e h a v e A (x) = x^{x - 1} = e^{(x - 1) l n (x)} = e^{x l n (x) - l n (x)} \to e^{+ \infty} = + \infty

(x \to 0^{+})

Commented by kaivan.ahmadi last updated on 19/May/19

{l i m}_{x \to 0} x^{x - 1} = {(0)}^{- 1} = + \infty

Commented by maxmathsup by imad last updated on 17/May/19

s i r x^{x} i s d e f i n e d o n] 0, + \infty [

Commented by kaivan.ahmadi last updated on 19/May/19

t h a n k u, i s i t t r u e n o w ?