find-lim-x-a-x-a-a-x-x-a-with-a-gt-0-

Question Number 49927 by Abdo msup. last updated on 12/Dec/18

f i n d {l i m}_{x \to a^{+}} (x - a) (a^{x} - x^{a}) w i t h a > 0 .

Commented by afachri last updated on 12/Dec/18

i found 0, Sir .

Commented by mathmax by abdo last updated on 27/Feb/20

l e t f (x) = (x - a) (a^{x} - x^{a}) c h a n g e m e n t x - a = t g i v e f (x) = g (t)

= t (a^{a + t} - {(a + t)}^{a}) a n d x \to a^{+} \Rightarrow t \to 0

g (t) = t {e^{(a + t) l n (a)} - e^{a l n (a + t)}} w e h a v e

e^{a l n (a + t)} = e^{a l n a + a l n (1 + \frac{t}{a})} \sim e^{a l n a + t} \Rightarrow

g (t) \sim t e^{a l n a} {1 - e^{t}} a n d e^{t} \sim 1 + t \Rightarrow g (t) \sim - t^{2} e^{a o n a} \to 0 (t \to 0) \Rightarrow

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