<s>[INST] Steps to solve $\lim_{n \to \infty} (\frac{ \sqrt{n^4+1} - \sqrt{n^4-1}}{ \frac1{(2n+1)^2}} ) = 4 $ $$\lim_{n \to \infty} \left(\frac{ \sqrt{n^4+1} - \sqrt{n^4-1}}{ \frac1{(2n+1)^2}} \right) = 4 $$
I think $\sqrt{n^4+1} - \sqrt{n^4-1}$ is approaching to zero, but it is not correct. What steps can evaluate above limit to 4?
 [/INST] You could also write $\sqrt{n^4 \pm 1} = n^2 \sqrt{1 \pm \frac{1}{n^4}}$ and expand the square roots out using the Binomial Theorem (for large $n, \, \frac{1}{n^4}$ is small), and see what this leaves you. 
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