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Question
Examine the continuity of the following function at given point:
`f(x)=(logx-log8)/(x-8) , `
` =8, `
Solution
Given f(8)=8
`lim_(x->0)f(x)=lim_(x->0)(logx-log8)/(x-8)`
`=Put x-8=h " then " x=8+h`
`x->8,h->8`
`f(8+h)=lim_(x->0)(log(8+h)-log8)/((8+h)-8)`
`=lim_(x->0)(log((h+8)/8))/h`
`=lim_(x->0)1/hlog(h+8)/8`
`=lim_(x->0)log[((h+8)/8)^(8/h)]^(1/8)`
`therefore f(x)=[1/8loge]=1/8nef(8)`
since `lim_(x->8) f(x)ne f(8) ` is discontinuous at x =8
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