Evaluate-lim-x-0-x-m-log-x-n-m-n-N-

Question Number 34464 by rahul 19 last updated on 06/May/18

E v a l u a t e

\lim_{x \to 0^{+}} x^{m} {(l o g x)}^{n}, m, n \in N

Answered by MJS last updated on 07/May/18

x^{m} {(\ln x)}^{n} = {(x^{\frac{m}{n}} \ln x)}^{n} = {(\frac{\ln x}{x^{- \frac{m}{n}}})}^{n}

l^{'} Hoptial :

\lim_{x \to 0^{+}} \frac{\ln x}{x^{- \frac{m}{n}}} = \lim_{x \to 0^{+}} \frac{\frac{1}{x}}{- \frac{m}{n} x^{- \frac{m}{n} - 1}} = \lim_{x \to 0^{+}} - \frac{n}{m} x^{\frac{m}{n}} = 0 \Rightarrow

\Rightarrow \lim_{x \to 0^{+}} x^{m} {(\ln x)}^{n} = 0

Commented by rahul 19 last updated on 07/May/18

I s t h i s m e t h o d c o r r e c t ?

Y o u a r e a p p l y i n g L - h o p i t a l a t

\frac{\ln x}{x^{\frac{- m}{n}}} b u t t h e o r i g i n a l e x p r e s s i o n t o

w h i c h w e s h o u l d a p p l y i s

{(\frac{l n x}{x^{\frac{- m}{n}}})}^{n}

!!!!

Commented by MJS last updated on 07/May/18

I think {it}^{'} s correct because if the limit of

f (x) is l the limit of {(f (x))}^{n} with n \in N should

be l^{n} \dots

Commented by rahul 19 last updated on 07/May/18

A b d o, T a n m a y s i r, c a n a n y o n e p l s

c o n f i r m t h e a b o v e s t a t e m e n t .

(I s i t a l w a y s t r u e ?)

I h a v e h e a r d t h a t :

\lim_{x \to 0} f (g (x)) = f (\lim_{x \to 0} g (x)) o n l y w h e n

f i s c o n t i n o u s a t t h a t p o i n t .

Commented by math khazana by abdo last updated on 08/May/18

y e s s i r r a h u l f m u s t b e c o n t i n u e t o a p l l y t h i s

r e s u l t \dots