Question

Examine the continuity of the following function at given point:

`f(x)=(logx-log8)/(x-8) , `

` =8, `

Answer 1

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