Find the value of `k ,` for which `f(x)={((sqrt(1+k x)-\ sqrt(1-k x))/(x) , if -1lt=x<0),( (2x+1)/(x-1), 0lt=x<1):}` is continuous at `x=0`

Explanation

The given function `f(x)` is continuous at `x=0`.



`lim _{x -> 0} f(x) `


`frac{0+1}{0-1}=lim _{x -> 0}(frac{sqrt{1+k x}-sqrt{1-k x}}{x}) `


`Rightarrow-1=lim _{x -> 0}(frac{sqrt{1+k x}-sqrt{1-k x}}{x})(frac{sqrt{1+k x}+sqrt{1-k x}}{sqrt{1+k x}+sqrt{1-k x}}) `


`Rightarrow-1=lim _{x -> 0}(frac{1+{kx}-1+{kx}}{x[sqrt{1+k x}+sqrt{1-k x}]}) `


`Rightarrow-1=lim _{x -> 0}(frac{2 k}{sqrt{1+k x}+sqrt{1-k x}}) `


`text { When } k=-1 `


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