f-x-0-and-lim-x-a-f-x-0-lim-x-a-g-x-then-lim-x-a-f-x-g-x-

Question Number 72905 by Tony Lin last updated on 04/Nov/19

f (x) ⩾ 0, a n d \lim_{x \to a} f (x) = 0, \lim_{x \to a} g (x) = \infty

t h e n \lim_{x \to a} f {(x)}^{g (x)} = ?

Commented by mathmax by abdo last updated on 04/Nov/19

w e h a v e b y t a y l o r f (x) = f (a) + (x - a) f^{^{'}} (a) + o (x - a)

\Rightarrow f (x) \sim (x - a) f^{^{'}} (a) b e c a u s e {l i m}_{x \to a} f (x) = 0

(w e s u p p o s e f d e r i v a b l e a t p o i n t a) \Rightarrow

f {(x)}^{g (x)} \sim {(x - a) f^{^{'}} (a)}^{g (x)} = e^{g (x) l n {(x - a) f^{^{'}} (a)}}

i f {l i m}_{x \to a} g (x) = + \infty w e h a v e {l i m}_{x \to a^{+}} l n {(x - a) f^{^{'}} (a)} = - \infty \Rightarrow

{l i m}_{x \to a^{+}} g (x) l n {(x - a) f^{^{'}} (a)} = - \infty \Rightarrow

{l i m}_{x \to a^{+}} {f (x)}^{g (x)} = 0

i f {l i m}_{x \to a^{+}} g (x) = - \infty \Rightarrow {l i m}_{x \to a^{+}} g (x) l n {(x - a) f^{^{'}} (a)} = + \infty \Rightarrow

{l i m}_{x \to a^{+}} {f (x)}^{g (x)} = + \infty a n d i f y o u h a v e a c l e a r a n s w e r p o s t i t . .

Commented by Tony Lin last updated on 05/Nov/19

t h a n k s s i r

Commented by mathmax by abdo last updated on 05/Nov/19

y o u a r e w e l c o m e .

Answered by mind is power last updated on 04/Nov/19

f {(x)}^{g (x)} = e^{g (x) \ln (f (x))}

\lim g (x) = + \infty

\lim [f (x) \to 0

\Rightarrow \lim \ln (f (x)) \to - \infty

\Rightarrow g (x) \ln (f) \to - \infty

\Rightarrow e^{g (x) \ln (f (x))} \to 0

Commented by Tony Lin last updated on 05/Nov/19

t h a n k s s i r

Answered by Tony Lin last updated on 05/Nov/19

\lim_{x \to a} f {(x)}^{g (x)} = 0

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