Given #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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Feb 9, 2018

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# \ #

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

Explanation:

# \ #

# "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. #

# \ #

# "Thus:" #

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

# \ #

# "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} }. #

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