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calculatelim-x-0-x-sinx-sinx-x-x-




Question Number 32517 by abdo imad last updated on 26/Mar/18
calculatelim_(x→0^+ )   ((x^(sinx)   −(sinx)^x )/x) .
$${calculatelim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\frac{{x}^{{sinx}} \:\:−\left({sinx}\right)^{{x}} }{{x}}\:. \\ $$
Commented by abdo imad last updated on 27/Mar/18
we have x^(sinx)  = e^(sinx ln(x))  but sinx ∼ x −(x^3 /6)  e^(sinxln(x))  ∼ e^((x−(x^3 /6))lnx)  ∼ 1+(x−(x^3 /6))lnx  (sinx)^x  =e^(xln(sinx))  ∼ e^(xln(x−(x^3 /3)))  =e^(xlnx +xln(1−(x^2 /3)))   ∼ e^(xlnx)  e^(−(x^3 /3))  ∼ e^(xlnx)  (1−(x^3 /3))∼ ⇒  ((x^(sinx)  −(sinx)^x )/x)  ∼  ((1+(x−(x^3 /6))lnx −e^(xlnx)  +(x^3 /3) e^(xlnx) )/x)  ∼ (x^2 /3) e_(x→0^+ ) ^(xln(x))  →0  .
$${we}\:{have}\:{x}^{{sinx}} \:=\:{e}^{{sinx}\:{ln}\left({x}\right)} \:{but}\:{sinx}\:\sim\:{x}\:−\frac{{x}^{\mathrm{3}} }{\mathrm{6}} \\ $$$${e}^{{sinxln}\left({x}\right)} \:\sim\:{e}^{\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\right){lnx}} \:\sim\:\mathrm{1}+\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\right){lnx} \\ $$$$\left({sinx}\right)^{{x}} \:={e}^{{xln}\left({sinx}\right)} \:\sim\:{e}^{{xln}\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\right)} \:={e}^{{xlnx}\:+{xln}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{3}}\right)} \\ $$$$\sim\:{e}^{{xlnx}} \:{e}^{−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}} \:\sim\:{e}^{{xlnx}} \:\left(\mathrm{1}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\right)\sim\:\Rightarrow \\ $$$$\frac{{x}^{{sinx}} \:−\left({sinx}\right)^{{x}} }{{x}}\:\:\sim\:\:\frac{\mathrm{1}+\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\right){lnx}\:−{e}^{{xlnx}} \:+\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\:{e}^{{xlnx}} }{{x}} \\ $$$$\sim\:\frac{{x}^{\mathrm{2}} }{\mathrm{3}}\:{e}_{{x}\rightarrow\mathrm{0}^{+} } ^{{xln}\left({x}\right)} \:\rightarrow\mathrm{0}\:\:. \\ $$

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