lim_(x→a) ((a^x −a^n )/(nln(x)−nln(a)))=?


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Question Number 168821 by mathlove last updated on 18/Apr/22

$\lim_{x \to a} \frac{a^{x} - a^{n}}{n l n (x) - n l n (a)} = ?$
	Answered by alephzero last updated on 18/Apr/22

	$\lim_{x \to a} \frac{a^{x} - a^{n}}{n \ln x - n \ln a} =$ $= \lim_{x \to a} \frac{a^{x} \ln a}{\frac{n}{x}} = \frac{a^{a + 1} \ln a}{n}$
	Commented by aleks041103 last updated on 18/Apr/22

	${Y o u}^{'} r e u s i n g l^{'} {h o p i t a l}^{'} s r u l e, b u t t o d o s o$ $t h e l i m i t m u s t b e o f t h e f o r m [\frac{0}{0}] o r [\frac{\infty}{\infty}] .$ $F o r t h i s l i m i t t h e d e n o m i n a t o r d o e s g o t o 0,$ $b u t t h e n u m e r a t o r g o e s t o 0 o n l y i f n = a .$ $I f a = n, t h e n y o u r a n s w e r i s c o r r e c t,$ $o t h e r w i s e t h e l i m i t g o e s t o \pm \infty d e p e n d i n g$ $o n w h i c h i s b i g g e r a^{a} o r a^{n} .$
	Commented by alephzero last updated on 19/Apr/22

	$T h a n k y o u s i r! I^{'} l l k n o w$
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