$\lim _{x \rightarrow \infty} x^3\left\{\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right\}=$

$\lim _{x \rightarrow \infty} x^3\left\{\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right\}=$
  1. $\frac{1}{\sqrt{2}}$
  2. $\frac{1}{4 \sqrt{2}}$
  3. $\frac{-1}{4 \sqrt{2}}$
  4. $\frac{-1}{\sqrt{2}}$

Solution

$\begin{aligned} & \lim _{x \rightarrow \infty} x^3\left(\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right) \\ & =\lim _{x \rightarrow \infty} \frac{x^3\left(x^2+\sqrt{1+x^4}-2 x^2\right)}{\sqrt{x^2+\sqrt{1+x^4}}+x \sqrt{2}} \\ & =\lim _{x \rightarrow \infty} \frac{x^3\left(\sqrt{1+x^4}-x^2\right)}{\sqrt{x^2+\sqrt{1+x^4}}+x \sqrt{2}} \\ & =\lim _{x \rightarrow \infty} \frac{x^3\left(1+x^4-x^4\right)}{\left(\sqrt{x^2+\sqrt{1+x^4}}+x \sqrt{2}\right)\left(\sqrt{1+x^4}+x^2\right)}\end{aligned}$ $\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{x^3}{x^3\left(\sqrt{1+\sqrt{\frac{1}{x^4}+1}+\sqrt{2}}\right)\left(\sqrt{\frac{1}{x^4}+1}+1\right)} \\ & =\lim _{x \rightarrow \infty} \frac{1}{\left(\sqrt{1+\sqrt{\frac{1}{x^4}+1}+\sqrt{2}}\right)\left(\sqrt{\frac{1}{x^4}+1}+1\right)} \\ & =\frac{1}{(\sqrt{1+1}+\sqrt{2})(1+1)} \\ & =\frac{1}{4 \sqrt{2}}\end{aligned}$

Asked in: MHT CET 2023 (14 May Shift 1)

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