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Lim(x->oo)((cosh(pi/x))/(cos(pi/x)))^(x^...

`Lim_(x->oo)((cosh(pi/x))/(cos(pi/x)))^(x^2)` where `cosh t=(e^t+e^-t)/2`

Text Solution

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`lim_(x->oo)((e^(pi/x)+e^(-pi/x))/(2*cos(pi/x)))*x^2`
`lim_(x->oo)x^2((e^(pi/x)+e^(-pi/x))/(2cos(pi/x))-1)`
`e^x=1+x+x^2/(2!)+x^3/(3!)..`
`e^(pi/x)=1+pi/x+((pi/x)^2/(2!))+...`
`e^(-pi/x)=1-pi/x+((pi/x)^2/(2!))...`
`(e^(pi/x)+e^(-pi/x))/2=2/2+((pi/x)^2/2)+2*(((pi/x)^4)/(4!))...`
`(e^(pi/x)+e^(-pi/x))/2=+((pi/x)^2/2)+(pi/x)^4+..`
`e^(lim_(x->oo))x^2(1+(pi/x)^2+...)/(cospi/x)`
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