Need help expanding this negative binomial expanison

Question

I am an A level Student studying for my papers and this is a question from a past paper...the original equation is $$\frac{1}{\sqrt{1+x}+\sqrt{1-x}}$$ which part a says to convert to the following equation and than says to expand it till x^2 $$\frac{\sqrt{1+x}-\sqrt{1-x}}{2x}$$ need to expand this equation till x^2 and have been doing for a long time but cant ..need to do it quick...thank you for the help

Answer 1

Using the binomial expansion, $$\sqrt{1+x}=1+\dfrac12x-\dfrac18x^2+\dfrac1{16}x^3+o(x^4)$$ and $$\sqrt{1-x}=1-\dfrac12x-\dfrac18x^2-\dfrac1{16}x^3+o(x^4)$$ so $$\dfrac{\sqrt{1+x}-\sqrt{1-x}}{2x}=\dfrac{x+\dfrac18x^3+o(x^4)}{2x}=\dfrac12+\dfrac1{16}x^2+o(x^3).$$