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Discuss the conjinuity of the functions...

Discuss the conjinuity of the functions at the points shown against them. If a function is discontinuous, determine whether the discontinuity is removable. In this case, redefine the function, so that it becomes continuous : `f(x)=log(100(0.01+x))/(3x), {:("for"x ne0),(",""for"x=0):}}at=0. =100/3`

Text Solution

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f(0)=100/3` " "` (Given)...(1)
`underset(xto0)(lim)f(x)=underset(xto0)lim(log100+(0.01+x))/(3x)`
`underset(x to0)(lim)(log(1+100x))/(3x)`
`=underset(xto0)(lim)(log(1+100x))/(100x)xx100/3`
`=100/3underset(xto0)lim(log(1+100x))/(100x)`
`=100/3xx1" "...[x to0, 100xto0and underset(t to0)lim(log(1+t))/(t)=1]`
`100/3" "...(2)`
From (1) and (2) `underset(x to 0)limf(x)=f(0)`
`therefore f` is continous at `x=0.`
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