Prove the following

Given $\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{x^{2}}$

Now to rationalize the denominator by multiplying the given equation by its rationalizing factor we get

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{3}}-\sqrt{1-x^{3}}}{x^{2}}=\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{3}}-\sqrt{1-x^{3}}}{x^{2}} \times\left(\frac{\sqrt{1+x^{3}}+\sqrt{1-x^{3}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}\right)$

On simplifying the above equation we get

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{3}}-\sqrt{1-x^{3}}}{x^{2}} \times\left(\frac{\sqrt{1+x^{3}}+\sqrt{1-x^{3}}}{\sqrt{1+x^{3}}+\sqrt{1-x^{3}}}\right)=\lim _{x \rightarrow 0} \frac{\left(1+x^{2}\right)-\left(1-x^{3}\right)}{x^{2}\left(\sqrt{1+x^{3}}+\sqrt{1-x^{3}}\right)}$

The above equation can be written as

$\lim _{x \rightarrow 0} \frac{\left(1+x^{2}\right)-\left(1-x^{2}\right)}{x^{2}\left(\sqrt{1+x^{3}}+\sqrt{1-x^{3}}\right)}=\lim _{x \rightarrow 0} \frac{2 x^{2}}{x^{2}\left(\sqrt{1+x^{3}}+\sqrt{1-x^{3}}\right)}=\lim _{x \rightarrow 0} \frac{2 x}{\left(\sqrt{1+x^{3}}+\sqrt{1-x^{3}}\right)}$

Now by applying the limit we get

$\Rightarrow$$\lim _{x \rightarrow 0} \frac{2 x}{\left(\sqrt{1+x^{2}}+\sqrt{1-x^{2}}\right)}=0$

$\Rightarrow$$\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{3}}-\sqrt{1-x^{2}}}{x^{2}}=0$