lim_(x→+∞) x{e−(1+(1/x))^x }=?


Question and Answers Forum

All Questions Topic List
Limits Questions
Previous in All Question Next in All Question
Previous in Limits Next in Limits
Question Number 79236 by jagoll last updated on 23/Jan/20
$lim_(x→+∞) x{e−(1+(1/x))^x }=?$
$\lim_{x \to + \infty} x {e - {(1 + \frac{1}{x})}^{x}} = ?$
Commented by mathmax by abdo last updated on 23/Jan/20
$let A(x)=x{e−(1+(1/x))^x } we have (1+(1/x))^x =e^(xln(1+(1/x))) we have ln^′ (1+u) =(1/(1+u)) =1−u +o(u^2 ) ⇒ ln(1+u) =u−(u^2 /2) +o(u^3 )(u∼o) ⇒ln(1+(1/x))=(1/x)−(1/(2x^2 )) +o((1/x^3 ))(x∼+∞) ⇒ xln(1+(1/x))=1−(1/(2x)) +o((1/x^2 )) ⇒e^(xln(1+(1/x))) =e^(1−(1/(2x))+o((1/x^2 ))) =e(1−(1/(2x)) +o((1/x^2 ))) ⇒e−(1+(1/x))^x =(e/(2x)) +o((1/x^2 )) ⇒ x{e−(1+(1/x))^x } =(e/2) +o((1/x)) ⇒lim_(x→+∞) A(x)=(e/2)$
$l e t A (x) = x {e - {(1 + \frac{1}{x})}^{x}} w e h a v e {(1 + \frac{1}{x})}^{x} = e^{x l n (1 + \frac{1}{x})}$ $w e h a v e {l n}^{^{'}} (1 + u) = \frac{1}{1 + u} = 1 - u + o (u^{2}) \Rightarrow$ $l n (1 + u) = u - \frac{u^{2}}{2} + o (u^{3}) (u \sim o) \Rightarrow l n (1 + \frac{1}{x}) = \frac{1}{x} - \frac{1}{2 x^{2}} + o (\frac{1}{x^{3}}) (x \sim + \infty) \Rightarrow$ $x l n (1 + \frac{1}{x}) = 1 - \frac{1}{2 x} + o (\frac{1}{x^{2}}) \Rightarrow e^{x l n (1 + \frac{1}{x})} = e^{1 - \frac{1}{2 x} + o (\frac{1}{x^{2}})}$ $= e (1 - \frac{1}{2 x} + o (\frac{1}{x^{2}})) \Rightarrow e - {(1 + \frac{1}{x})}^{x} = \frac{e}{2 x} + o (\frac{1}{x^{2}}) \Rightarrow$ $x {e - {(1 + \frac{1}{x})}^{x}} = \frac{e}{2} + o (\frac{1}{x}) \Rightarrow {l i m}_{x \to + \infty} A (x) = \frac{e}{2}$
Commented by jagoll last updated on 23/Jan/20

$thank you sir$
Commented by mathmax by abdo last updated on 24/Jan/20

$y o u a r e w e l c o m e .$
	Answered by Smail last updated on 23/Jan/20

	$x = 1 / t$ $\underset{t \to 0}{l i m} \frac{e - {(1 + t)}^{1 / t}}{t} = \underset{t \to 0}{l i m} \frac{e - e^{\frac{l n (1 + t)}{t}}}{t}$ $\underset{t \to 0}{l i m} \frac{e - e^{(t - \frac{t^{2}}{2}) \frac{1}{t}}}{t} = \underset{t \to 0}{l i m} \frac{e - e (1 - t / 2)}{t} = \frac{e}{2}$
	Commented by jagoll last updated on 23/Jan/20

	$thank you sir$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum