Let `f(x)=(log(1+x/a)-log(1-x/b))/x ,x!=0.` Find the value of `f` at `x=0` so that `f` becomes continuous at `x=0`

Explanation

`f(x) = (log(1) - log(1))/0 = 0/0`

`f(0) = f(0+h) = f(0-h) `

`l = lim_(x->0) f(x) = lim(x->0) (log(1 + x/a) - log(1-x/b))/x = 0/0`

applying l hospital rule

`l = lim_(x->0) (1/(1+ x/a) *1/a - 1/(1-x/b) (-1/b))/1`

`l = lim_(x->0) ((1/(1+0) *1/a + (1/(1-0) *1/b))/1)`

`l = 1/a + 1/b = (a+b)/(ab)`

so, `f(0) = (a+b)/(ab)`

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