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Question-94730




Question Number 94730 by i jagooll last updated on 20/May/20
Answered by john santu last updated on 20/May/20
lim_(x→0^+ )  e^(ln(x)^(sin x) ) = lim_(x→0)  e^(sin x. ln(x))   e^(lim_(x→0^+ )  (sin x).ln(x)) = e^(lim_(x→0^+ )  ((ln(x))/(csc x)))   e^(lim_(x→0^+ )  ((((1/x)))/(csc(x).cot (x)))) = e^(lim_(x→0^+ )  (((((sin x)/x)))/(cot (x))))   e^(lim_(x→0^+ )  ((tan (x).sin (x))/x))  = e^0  = 1
$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\mathrm{e}^{\mathrm{ln}\left(\mathrm{x}\right)^{\mathrm{sin}\:\mathrm{x}} } =\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{e}^{\mathrm{sin}\:\mathrm{x}.\:\mathrm{ln}\left(\mathrm{x}\right)} \\ $$$$\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left(\mathrm{sin}\:\mathrm{x}\right).\mathrm{ln}\left(\mathrm{x}\right)} =\:\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{csc}\:\mathrm{x}}} \\ $$$$\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\left(\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{csc}\left(\mathrm{x}\right).\mathrm{cot}\:\left(\mathrm{x}\right)}} =\:\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\left(\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\right)}{\mathrm{cot}\:\left(\mathrm{x}\right)}} \\ $$$$\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{tan}\:\left(\mathrm{x}\right).\mathrm{sin}\:\left(\mathrm{x}\right)}{\mathrm{x}}} \:=\:\mathrm{e}^{\mathrm{0}} \:=\:\mathrm{1}\: \\ $$
Commented by john santu last updated on 20/May/20
correct?
$$\mathrm{correct}? \\ $$
Commented by i jagooll last updated on 21/May/20
yes..correct
$$\mathrm{yes}..\mathrm{correct} \\ $$
Answered by mathmax by abdo last updated on 21/May/20
let f(x) =x^(sinx)  ⇒f(x) =e^(sinxln(x))   we have sinx lnx =((sinx)/x)×(xlnx) ⇒lim_(x→0^+ )  sinx lnx  =lim_(x→0^+ )   ((sinx)/x)(xlnx) =1×0 =0 ⇒lim_(x→0^+ )   f(x) =1
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{x}^{\mathrm{sinx}} \:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{e}^{\mathrm{sinxln}\left(\mathrm{x}\right)} \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{sinx}\:\mathrm{lnx}\:=\frac{\mathrm{sinx}}{\mathrm{x}}×\left(\mathrm{xlnx}\right)\:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}^{+} } \:\mathrm{sinx}\:\mathrm{lnx} \\ $$$$=\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}^{+} } \:\:\frac{\mathrm{sinx}}{\mathrm{x}}\left(\mathrm{xlnx}\right)\:=\mathrm{1}×\mathrm{0}\:=\mathrm{0}\:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}^{+} } \:\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{1} \\ $$

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