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Question Number 131052 by benjo_mathlover last updated on 01/Feb/21

lim_(x→s) ((x^s^s −s^x^s )/(x^2 −s^2 )) =?

$$\:\underset{{x}\rightarrow{s}} {\mathrm{lim}}\:\frac{{x}^{{s}^{{s}} } −{s}^{{x}^{{s}} } }{{x}^{\mathrm{2}} −{s}^{\mathrm{2}} }\:=? \\ $$

Answered by Ar Brandon last updated on 01/Feb/21

L=((s^s^s −s^s^s )/(s^2 −s^2 ))=(((s−s)Σ_(k=0) ^(s^s −1) s^(s^s −k−1) ∙s^k )/((s−s)(s+s))) =(1/(2s))Σ_(k=0) ^(s^s −1) s^(s^s −1) =(((s^(s^s −1) )s^s )/(2s))

$$\mathscr{L}=\frac{\mathrm{s}^{\mathrm{s}^{\mathrm{s}} } −\mathrm{s}^{\mathrm{s}^{\mathrm{s}} } }{\mathrm{s}^{\mathrm{2}} −\mathrm{s}^{\mathrm{2}} }=\frac{\left(\mathrm{s}−\mathrm{s}\right)\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{s}^{\mathrm{s}} −\mathrm{1}} {\sum}}\mathrm{s}^{\mathrm{s}^{\mathrm{s}} −\mathrm{k}−\mathrm{1}} \centerdot\mathrm{s}^{\mathrm{k}} }{\left(\mathrm{s}−\mathrm{s}\right)\left(\mathrm{s}+\mathrm{s}\right)} \\ $$$$\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2s}}\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{s}^{\mathrm{s}} −\mathrm{1}} {\sum}}\mathrm{s}^{\mathrm{s}^{\mathrm{s}} −\mathrm{1}} =\frac{\left(\mathrm{s}^{\mathrm{s}^{\mathrm{s}} −\mathrm{1}} \right)\mathrm{s}^{\mathrm{s}} }{\mathrm{2s}} \\ $$

Commented by Ar Brandon last updated on 01/Feb/21

a^n −b^n =(a−b)Σ_(k=0) ^(n−1) a^(n−k−1) b^k a=s=b, n=s^s

$$\mathrm{a}^{\mathrm{n}} −\mathrm{b}^{\mathrm{n}} =\left(\mathrm{a}−\mathrm{b}\right)\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\mathrm{a}^{\mathrm{n}−\mathrm{k}−\mathrm{1}} \mathrm{b}^{\mathrm{k}} \\ $$$$\mathrm{a}=\mathrm{s}=\mathrm{b},\:\mathrm{n}=\mathrm{s}^{\mathrm{s}} \\ $$

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