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Discuss the continuity of the following functions at the points shown against them : `{:(f(x)=(1-sinx)/(((pi)/(2)-x)^(2))",", "for"x ne(pi)/(2)),(=3",","for"x=(pi)/(2)):}}at x=(pi)/(2)*` |
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Answer» R.H.L. `=lim_(hto0)f((pi)/(2)+h)=lim_(hto0)(1-sin((pi)/(2)+h))/(((pi)/(2)-(pi)/(2)-h)^(2))` `=lim_(hto0)(1-cosh)/((-h)^(2))` `=lim_(hto0)(1-1+2sin^(2)((h)/(2)))/((-h)^(2))` `=lim_(hto0)(2sin^(2)((h)/(2)))/(h^(2))` `=lim_(hto0)(2sin^(2)((h)/(2)))/(4((h)/(2))^(2))` `=(1)/(2)lim_(hto0)(("sin"(h)/(2))/((h)/(2)))^(2)` `=(1)/(2)xx1=(1)/(2)` L.H.L. `=lim_(hto0)f((pi)/(2)-h)` `lim_(hto0)(1-sin((pi)/(2)-h))/(((pi)/(2)-(pi)/(2)+h)^(2))` `=lim_(hto0)(1-cosh)/(h^(2))` `=(1)/(2)` `=(1)/(2)ne3` R.H.L=L.H.L `ne((pi)/(2))` So,function is discontinuous at `x=(pi)/(2)`. |
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