Evaluating $\lim_{x\to\infty} x^{7/2} \left[\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}\left(1-\frac1{8x^2}\right)\right] $ [closed]
Solution 1:
Here's an idea to get you started.
$$\lim_{x\to\infty} x^{7/2} \left[\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}\left(1-\frac1{8x^2}\right)\right] $$
$$=\lim_{x\to\infty} x^{4} \left[\sqrt{1+\frac1x}+\sqrt{1-\frac1x}-2\left(1-\frac1{8x^2}\right)\right]. $$
Now use the binomial series $\sqrt{1+\dfrac1x}=1+\dfrac1{2x}-\dfrac1{8x^2}+\dfrac1{16x^3}-\dfrac5{128x^4}+O\left(\left(\dfrac1x\right)^5\right)$
and an analogous one for $\sqrt{1-\dfrac1x}.\;$ Can you take it from here?