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calculate-lim-x-1-nx-n-1-n-1-x-n-1-x-1-2-without-hospital-rule-

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Question Number 79103 by mathmax by abdo last updated on 22/Jan/20
calculate lim_(x→1) ((nx^(n+1) −(n+1)x^n +1)/((x−1)^2 )) without hospital rule.
$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\frac{{nx}^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right){x}^{{n}} \:+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:\:{without}\:{hospital}\:{rule}. \\ $$
Commented by mathmax by abdo last updated on 24/Jan/20
let f(x)=((nx^(n+1) −(n+1)x^n +1)/((x−1)^2 )) changement x−1=t give f(x)=g(t) =((n(1+t)^(n+1) −(n+1)(1+t)^n +1)/t^2 ) x→1 ⇒ t→0 we know (1+t)^α ∼1+αt +((α(α−1))/2)t^2 ⇒ (1+t)^(n+1) ∼1+(n+1)t +(((n+1)n)/2)t^2 and (1+t)^n ∼1+nt +((n(n−1))/2)t^2 ⇒ g(t)∼((n +n(n+1)t +((n^2 (n+1))/2)t^2 −(n+1)−n(n+1)t−((n(n−1)(n+1))/2)t^2 +1)/t^2 ) ⇒g(t)∼(({((n^2 (n+1))/2)−((n(n−1)(n+1))/2)}t^2 )/t^2 ) ⇒ g(t) ∼((n(n+1))/2)(n−n+1) ⇒lim_(t→0) g(t)=((n(n+1))/2) ⇒ lim_(x→1) f(x)=((n(n+1))/2)
$${let}\:{f}\left({x}\right)=\frac{{nx}^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right){x}^{{n}} \:+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:{changement}\:{x}−\mathrm{1}={t}\:{give} \\ $$$${f}\left({x}\right)={g}\left({t}\right)\:=\frac{{n}\left(\mathrm{1}+{t}\right)^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right)\left(\mathrm{1}+{t}\right)^{{n}} \:+\mathrm{1}}{{t}^{\mathrm{2}} } \\ $$$${x}\rightarrow\mathrm{1}\:\Rightarrow\:{t}\rightarrow\mathrm{0}\:\:\:\:{we}\:{know}\:\:\left(\mathrm{1}+{t}\right)^{\alpha} \:\sim\mathrm{1}+\alpha{t}\:+\frac{\alpha\left(\alpha−\mathrm{1}\right)}{\mathrm{2}}{t}^{\mathrm{2}} \:\Rightarrow \\ $$$$\left(\mathrm{1}+{t}\right)^{{n}+\mathrm{1}} \:\sim\mathrm{1}+\left({n}+\mathrm{1}\right){t}\:+\frac{\left({n}+\mathrm{1}\right){n}}{\mathrm{2}}{t}^{\mathrm{2}} \:{and} \\ $$$$\left(\mathrm{1}+{t}\right)^{{n}} \:\sim\mathrm{1}+{nt}\:+\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}{t}^{\mathrm{2}} \:\Rightarrow \\ $$$${g}\left({t}\right)\sim\frac{{n}\:+{n}\left({n}+\mathrm{1}\right){t}\:+\frac{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)}{\mathrm{2}}{t}^{\mathrm{2}} −\left({n}+\mathrm{1}\right)−{n}\left({n}+\mathrm{1}\right){t}−\frac{{n}\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{2}}{t}^{\mathrm{2}} \:+\mathrm{1}}{{t}^{\mathrm{2}} } \\ $$$$\Rightarrow{g}\left({t}\right)\sim\frac{\left\{\frac{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)}{\mathrm{2}}−\frac{{n}\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{2}}\right\}{t}^{\mathrm{2}} }{{t}^{\mathrm{2}} }\:\Rightarrow \\ $$$${g}\left({t}\right)\:\sim\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\left({n}−{n}+\mathrm{1}\right)\:\Rightarrow{lim}_{{t}\rightarrow\mathrm{0}} \:\:{g}\left({t}\right)=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$ \\ $$

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