find lim_(x→a^+ ) (x−a)(a^x −x^a ) with a>0 .


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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 byafachri last updated on 12/Dec/18

$i found 0, Sir .$
Commented bymathmax by abdo last updated on 27/Feb/20
$let f(x)=(x−a)(a^x −x^a ) changement x−a=t givef(x)=g(t) =t(a^(a+t) −(a+t)^a ) and x→a^+ ⇒t→0 g(t) =t { e^((a+t)ln(a)) −e^(aln(a+t)) }we have e^(aln(a+t)) =e^(alna +aln(1+(t/a))) ∼e^(alna +t) ⇒ g(t)∼t e^(alna) {1−e^t } and e^t ∼1+t ⇒g(t)∼−t^2 e^(aona) →0 (t→0) ⇒ lim_(x→a^+ ) f(x)=0$
$l e t f (x) = (x - a) (a^{x} - x^{a}) c h a n g e m e n t x - a = t g i v e f (x) = g (t)$ $= t (a^{a + t} - {(a + t)}^{a}) a n d x \to a^{+} \Rightarrow t \to 0$ $g (t) = t {e^{(a + t) l n (a)} - e^{a l n (a + t)}} w e h a v e$ $e^{a l n (a + t)} = e^{a l n a + a l n (1 + \frac{t}{a})} \sim e^{a l n a + t} \Rightarrow$ $g (t) \sim t e^{a l n a} {1 - e^{t}} a n d e^{t} \sim 1 + t \Rightarrow g (t) \sim - t^{2} e^{a o n a} \to 0 (t \to 0) \Rightarrow$ ${l i m}_{x \to a^{+}} f (x) = 0$
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