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if f(x)= {cos²x -sin²x-1/under root of(x²-1)-1 ,x is not=0

{k ,x=0

is continuous at x=0 ,find k

$Since f(x) is continuous at x=0, ∴ \lim_{x \to 0} f (x) = f (0) \lim_{x \to 0} (\frac{\cos^{2} x - \sin^{2} x - 1}{\sqrt{x^{2} + 1} - 1}) = k \Rightarrow \lim_{x \to 0} (\frac{1 - \sin^{2} x - \sin^{2} x - 1}{\sqrt{x^{2} + 1} - 1}) = k \Rightarrow \lim_{x \to 0} (\frac{- {2 \sin}^{2} x}{\sqrt{x^{2} + 1} - 1}) = k \Rightarrow - 2 \lim_{x \to 0} \frac{\sin^{2} x}{\sqrt{x^{2} + 1} - 1} \times \frac{\sqrt{x^{2} + 1} + 1}{\sqrt{x^{2} + 1} + 1} = k \Rightarrow - 2 \lim_{x \to 0} (\frac{\sin x}{x})^{2} \times \lim_{x \to 0} \sqrt{x^{2} + 1} + 1 = k (∵ \lim_{x \to 0} \frac{\sin x}{x} = 1) \Rightarrow - 2 (1)^{2} \times (\sqrt{(0 + 1)} + 1) = k \Rightarrow - 2 \times 2 = k \Rightarrow k = - 4$

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