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Find that the function is continuous or discontinuous at the indicated point f(x) = {{:(|x-a|sin\ (1)/(x-a),if x ne a),(0, if x =a):} at x = a |
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Answer» At `LHL=UNDERSET(xrarra^(-))(lim)|x-a|sin'(1)/(x-a)` `=underset(hrarr0)(lim) |a-h-a| sin((h)/(a-h-a))` `= underset(hrarr0)(lim)-hsin(1/h) , [:' sin(-THETA)=-SINTHETA]` `=0 xx` [an oscillatingnumber between `-1` and 1] = 0 `RHL = underset (xrarra^(+))(lim)|x-a|sin((1)/(x-a))` `=underset(hrarr0)lim|a+h-a|sin((1)/(a+h-a))=underset(hrarr0)limh sin '1/h` `= 0xx` [ an oscillatingnumber between` -1` and 1] = 0 and `f(a) = 0` `:. LHL = RHL = f(a)` So, `f(x)` is continousat `x = a`. |
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