Lim (1-cos(1-sen(x)))/((pi/2)-x)^(2)as x->pi/2 ?

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Answer 1 · 2018-05-15T18:34:27

# 0#.

Suppose that, #x=pi/2+h#.

#:." As "x to pi/2, h to 0#.

#:." The Limit"=lim_(h to 0)(1-cos(1-sin(pi/2+h)))/(-h)^2#,

#=lim_(h to 0)(1-cos(1-cos h))/h^2#,

#=lim_(h to 0)(1-cos(2sin^2(h/2)))/h^2#,

#=lim_(h to 0)(2sin^2(sin^2(h/2)))/h^2...[because, 1-cos2y=2sin^2y]#,

#=lim_(h to 0)2{sin(sin^2(h/2))/h}^2#,

#=lim_(h to 0)2{sin(sin^2(h/2))/sin^2(h/2)*sin^2(h/2)/h}^2#,

#=lim2{sin(sin^2(h/2))/sin^2(h/2)*sin(h/2)/{(h/2)*2}*sin(h/2)}^2#,

#=lim2/2^2{sin(sin^2(h/2))/sin^2(h/2)*sin(h/2)/(h/2)*sin(h/2)}^2#.

# rArr" The Limit"=2/4{1*1*0}^2=0#.