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Question
Discuss the continuity of the function
`f(x)=(1-sinx)/(pi/2-x)^2, `
= 3, for x=π/2
Solution
`f(pi/2)=3 ......(given)`
`lim_(x->pi/2)f(x)=lim_(x->pi/2)(1-sinx)/(pi/2-x)`
`put pi/2-x=h then x=pi/2-h`
`As x->pi/2,h->0`
`lim_(h->0)(1-sin(pi/2-h))/h^2=lim_(h->0)(1-cosh)/h^2`
`=lim_(h->0)(1-cosh)/h^2xx(1+cosh)/(1+cosh)=lim_(h->0)(1-cos^2h)/(h^2(1+cosh))`
`=lim_(h->0)(sin^2h)/(h^2(1+cosh))=lim_(h->0)((sinh)/h)^2(1/(1+cosh))=1xx1/(1+1)`
`=lim_(x->pi/2)f(x)!=f(pi/2)`
f(x) is discontinuous at `x =pi/2`
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