If `f(x)={(cos^2x-s in^2x-1)/(sqrt(x^2+1)-1),\ \ \ x!=0k ,\ \ \ x=0` is continuous at `x=0` , find `kdot`

Explanation

`f(x)= frac{cos ^{2} x-sin ^{2} x-1}{sqrt{x^{2}+1}-1}, x neq 0 k, x=0` If `f(x)` is continuous at `x=0`, then


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


`Rightarrow lim _{x-> 0} frac{cos ^{2} x-sin ^{2} x-1}{sqrt{x^{2}+1}-1}=k`


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


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


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


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


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


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