Question

The value of $f\left(0\right)$ such $f\left(x\right)=\frac{1-{\cos }^{2}x+{\sin }^{2}x}{\sqrt{x^2+1}-1}(x\ne 0)$ is continuous at $x=0$ is

Answer 1

Given $f\left(x\right)=\frac{1-{\cos }^{2}x+{\sin }^{2}x}{\sqrt{x^2+1}-1}(x\ne 0)$
Since, $f\left(x\right)$ is continuous at $x=0$
${lim}_{x\to 0}f\left(x\right)=f\left(0\right)$
${lim}_{x\to 0}\frac{1-co{s}^{2}x+si{n}^{2}x}{\sqrt{x^2+1}-1}$
$=\underset{x\to 0}{lim}\frac{2si{n}^{2}x}{x^2}(\sqrt{x^2+1}+1)$
$=2{lim}_{x\to 0}{\left(\frac{sinx}{x}\right)}^2{lim}_{x\to 0}(\sqrt{x^2+1}+1)$
$=2\times 1\times 2=4$
$\therefore f\left(0\right)=4$