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Discuss the continuity of the following ...

Discuss the continuity of the following functions at the points shown against them :
`{:(f(x)=(logx-log8)/(x-8)",","for"x ne0),(=10/7",","for"x=0):}}at x=8.`

Text Solution

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`f(8)=8" "..."(Given")...(1)`
`underset(xto8)limf(x)=underset(xto8)lim(logx-log8)/(x-8)`
Put `x=8` Then `x-8=h` and as `x to8, hto 0.`
`underset(xto8)limf(x)=underset(xto0)lim(log(8+h)-log8)/(h)`
`=underset(hto0)lim(log((8+h)/(8)))/(((h)/(8)))xx1/8`
`=1/8underset(hto0)lim(log(1+(h)/(8)))/(((h)/(8)))`
`=1/8(1)...[hto0,h/8to0and underset(xto0)lim(log (1+x))/(x)=1=1/8`
From (1) and (2), `underset(xto8)lim f(x)nef(8)`
`therefore f` is discontanous at `x=8.`
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