$\lim_{x->0}\frac{\sqrt{1+x}-\sqrt{1-x}}{x}$ [putting x=0, it comes 0/0 form , So, apply L hospital rule(differentiate upper limit and lower limit differently)]
$\lim _{x \rightarrow 0}\frac{1}{2\sqrt{1+x}}+\frac{1}{2\sqrt{1-x}}=\frac{1}{2}+\frac{1}{2}=1$