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Question Number 40759 by math khazana by abdo last updated on 27/Jul/18

calculate lim_(n→+∞)      ((1−e^(−nx^2 ) )/(x^2 sin((π/n))))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:\:\frac{\mathrm{1}−{e}^{−{nx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} {sin}\left(\frac{\pi}{{n}}\right)} \\ $$

Commented by math khazana by abdo last updated on 27/Jul/18

the Q is find lim_(x→0)    ((1−e^(−nx^2 ) )/(x^2  sin(((πx)/n))))

$${the}\:{Q}\:{is}\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{\mathrm{1}−{e}^{−{nx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} \:{sin}\left(\frac{\pi{x}}{{n}}\right)} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 27/Jul/18

lim_(x→0 )  ((e^(nx^2 ) −1)/(nx^2 ))×(n/e^(nx^2 ) )×(1/({((sin(((Πx)/n)))/((Πx)/n))}))×(1/((Πx)/n))  =1×(n/1)×(1/1)×(n/0)=∞

$${li}\underset{{x}\rightarrow\mathrm{0}\:} {{m}}\:\frac{{e}^{{nx}^{\mathrm{2}} } −\mathrm{1}}{{nx}^{\mathrm{2}} }×\frac{{n}}{{e}^{{nx}^{\mathrm{2}} } }×\frac{\mathrm{1}}{\left\{\frac{{sin}\left(\frac{\Pi{x}}{{n}}\right)}{\frac{\Pi{x}}{{n}}}\right\}}×\frac{\mathrm{1}}{\frac{\Pi{x}}{{n}}} \\ $$$$=\mathrm{1}×\frac{{n}}{\mathrm{1}}×\frac{\mathrm{1}}{\mathrm{1}}×\frac{{n}}{\mathrm{0}}=\infty \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Jul/18

lim_(n→∞)   ((1−(1/e^(nx^2 ) ))/(x^2 sin((Π/n))))  when  n→∞  (1/e^(nx^2 ) ) →0   and for any value of  n  the value of sin((Π/n)) lies between ±1          (1/x^2 )>  lim_(n→∞)   ((1−e^(−nx^2 ) )/(x^2 sin((Π/n))))>−(1/x^2 )                        li_(t→0)

$${li}\underset{{n}\rightarrow\infty} {{m}}\:\:\frac{\mathrm{1}−\frac{\mathrm{1}}{{e}^{{nx}^{\mathrm{2}} } }}{{x}^{\mathrm{2}} {sin}\left(\frac{\Pi}{{n}}\right)} \\ $$$${when}\:\:{n}\rightarrow\infty\:\:\frac{\mathrm{1}}{{e}^{{nx}^{\mathrm{2}} } }\:\rightarrow\mathrm{0}\:\:\:{and}\:{for}\:{any}\:{value}\:{of} \\ $$$${n}\:\:{the}\:{value}\:{of}\:{sin}\left(\frac{\Pi}{{n}}\right)\:{lies}\:{between}\:\pm\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }>\:\:{li}\underset{{n}\rightarrow\infty} {{m}}\:\:\frac{\mathrm{1}−{e}^{−{nx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} {sin}\left(\frac{\Pi}{{n}}\right)}>−\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{li}} \\ $$

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