Examine the contionuity of the following functions at the give points `f(x)={(log100+log(0. 01+x))/(3x)` for `x!=0` and `100/3` for x =0 } at `x=0`

Explanation

f (x ) is said to be continuous at x = 0 , if
`underset(x to 0) lim f (x) = f (0) ` ....(1)
Now ` f(0) = 100/3 `(given) ...(2)
` underset(x to 0) lim f(x) = underset(x to 0) lim (log 100 + log (0.01 + x))/(3x)`
` = underset(x to 0) lim (log (100 xx 0.01 + 100 x))/(3x) `
` = underset(x to 0) lim (log (1 + 100x))/(3x)`
` = 100/3 underset(x to 0) lim (log (1+100 x))/(100 x) `
` = 100/3 xx 1 = 100/3 ` ...(3)
` :.` From (2) and (3),
` underset(x to 0) lim f (x) = f(0) `
` :. ` f(x) is continuous at x = 0 .