Mathematics - Limits Question with Solution | TestHub

We have $f(x)=\frac{x^3\left\{x^2+\sqrt{x^4+1}-2x^2\right\}}{\sqrt{x^2+\sqrt{x^4+1}}+x\sqrt{2}}$

$$\begin{array}{rl}& =\frac{x^3\left\{\sqrt{x^4+1}-x^2\right\}}{\sqrt{x^2+\sqrt{x^4+1}}+x\sqrt{2}}\\ & =\frac{x^3\left(x^4+1-x^4\right)}{\left[\sqrt{x^2+\sqrt{x^4+1}}+x\sqrt{2}\right]\left[\sqrt{x^4+1}+x^2\right]}\\ & =\frac{x^3}{\left[\sqrt{x^2+\sqrt{x^4+1}}+x\sqrt{2}\right]\left[\sqrt{x^4+1}+x^2\right]}\\ & =\frac{1}{\left[\sqrt{1+\sqrt{1+\frac{1}{x^4}}}+\sqrt{2}\right]\left[\sqrt{1+\frac{1}{x^4}}+1\right]}\\ & =\frac{1}{(\sqrt{1+\sqrt{1}}+\sqrt{2})(\sqrt{1}+1)}\\ & =\frac{1}{2\sqrt{2}(2)}=\frac{1}{4\sqrt{2}}\end{array}$$