Prove that lim_(x→0) (((1+ax)^(1/b) −1)/x) = (a/b).


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Question Number 44652 by rahul 19 last updated on 02/Oct/18

$P r o v e t h a t \lim_{x \to 0} \frac{{(1 + a x)}^{\frac{1}{b}} - 1}{x} = \frac{a}{b} .$
Commented by rahul 19 last updated on 02/Oct/18

$w i t h o u t u s i n g L - h o s p i t a l r u l e .$
Commented by maxmathsup by imad last updated on 02/Oct/18

$w e h a v e {(1 + a x)}^{\frac{1}{b}} \sim 1 + \frac{a}{b} x (x \to 0) \Rightarrow$ ${(1 + a x)}^{\frac{1}{b}} - 1 \sim \frac{a}{b} x (x \to 0) \Rightarrow \frac{{(1 + a x)}^{\frac{1}{b}} - 1}{x} \sim \frac{a}{b} \Rightarrow$ ${l i m}_{x \to 0} \frac{{(1 + a x)}^{\frac{1}{b}} - 1}{x} = \frac{a}{b} .$
Commented by rahul 19 last updated on 02/Oct/18

$t h a n k s p r o f A b d o .$
Commented by maxmathsup by imad last updated on 02/Oct/18

$y o u a r e w e l c o m e s i r .$
	Answered by math1967 last updated on 02/Oct/18
	$x^(lim) →0 ((a{(1+ax)^(1/b) −1})/(ax)) a ^(lim) x→0 (((1+ax)^(1/b) −1)/(ax)) (a/b) proved [ ^(Lim) x→0 (((1+x)^n −1)/x) ( n=any rational..) =n]$
	$\overset{l i m}{x} \to 0 \frac{a {{(1 + a x)}^{\frac{1}{b}} - 1}}{a x}$ $a \overset{l i m}{} x \to 0 \frac{{(1 + a x)}^{\frac{1}{b}} - 1}{a x}$ $\frac{a}{b} p r o v e d [\overset{L i m}{} x \to 0 \frac{{(1 + x)}^{n} - 1}{x} (n = a n y r a t i o n a l . .)$ $= n]$
	Commented by rahul 19 last updated on 02/Oct/18

	$t h a n k s s i r .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 02/Oct/18

	$t = 1 + a x x \to 0 t \to 1$ $\lim_{t \to 1} \frac{a (t^{\frac{1}{b}} - 1)}{t - 1}$ $a \times \frac{1}{b} \times {(1)}^{\frac{1}{b} - 1}$ $= \frac{a}{b}$
	Commented by rahul 19 last updated on 02/Oct/18

	$t h a n k s s i r .$
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