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Examine the continuity of the following function at given point : `f(x)=(log x-log 8)/(x-8)", for "x ne 8` `"8 ,for "x=8` `"at , "x=8` |
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Answer» Given `f(8)=8" …(i)"` `underset(xrarr8)(lim)f(x)=underset(xrarr8)(lim)(logx-log8)/(x-8)` Putting `x=8+h,` then `x-8=h` and as `xrarr8, hrarr0.` `therefore" "underset(xrarr8)(lim)f(x)=underset(hrarr0)(lim)(log(8+h)-log8)/(h)` `=underset(hrarr0)(lim)(log((8+h)/(8)))/(h)` `=underset(hrarr0)(lim)(log(1+(h)/(8)))/((h)/(8))xx(1)/(8)` `=(1)/(8)xx1(because underset(xrarr0)(lim)(log(1+x))/(x)=1)" ...(ii)"` From equation (i) and (ii), `underset(xrarr8)(lim)f(x) ne f(8)` `therefore" f is discontinuous at x = 8."` |
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