Given $a = 1 + \sqrt{2}$ $a = 1+sqrt2$ find $lim_(x->0)((a+x)^a/a^(a+x))^(1/x)$ Try not to use the L'Hopital method.?

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Answer 1 · 2018-02-09T19:32:46

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$"Answer is:" \qquad e/{ 1 + \sqrt{2} }.$

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$"We compute as follows:"$

$lim_{x rarr 0} ( ( a + x )^a / a^{a +x} )^{1/x} \ = \ lim_{x rarr 0} ( 1/ a^x )^{1/x} \cdot ( ( a + x )^a / a^{a} )^{1/x}$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ lim_{x rarr 0} 1/ a \cdot ( ( { a + x }/a )^a)^{1/x}$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ lim_{x rarr 0} 1/ a \cdot ( ( 1 + x/a )^a)^{1/x}$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ lim_{x rarr 0} 1/ a \cdot ( 1 + x/a )^{a/x}$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = 1/ a \cdot e$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = e / a.$

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$"Thus:"$

$\qquad \qquad \qquad \qquad \qquad \qquad \quad \ lim_{x rarr 0} ( ( a + x )^a / a^{a +x} )^{1/x} \ = \ e / a.$

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$"In the case of this problem:" \quad a = 1 + \sqrt{2}.$

$\qquad \qquad \qquad \qquad \qquad :. \qquad \quad \ lim_{x rarr 0} ( ( a + x )^a / a^{a +x} )^{1/x} \ = \ e / { 1 + \sqrt{2} }.$

Given a = 1+sqrt2a=1+√2a = 1+sqrt2 find lim_(x->0)((a+x)^a/a^(a+x))^(1/x)lim_(x->0)((a+x)^a/a^(a+x))^(1/x) Try not to use the L'Hopital method.?