Question

Discuss the continuity of the function

`f(x)=(1-sinx)/(pi/2-x)^2, `

       = 3,                  for x=π/2

 

Answer 1

`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`