Question

$[tex][tex] \lim_{n \to \infty} \sqrt{2 n^{2} +5n+2}- \sqrt{an^{2} +bn+c}=5 \sqrt{2$

a,b,c=?

Answer 1

[tex] \lim_{n \to \infty} \frac{2n^2+5n+2-an^2-bn-c}{\sqrt{2n^2+5n+2}+\sqrt{an^2+bn+c}} =5\sqrt{2}\\ \lim_{n \to \infty} \frac{(2-a)n^2+(5-b)n+2-c}{n(\sqrt{2+5/n+2/n^2}+\sqrt{a+b/n+c/n^2}} =5\sqrt{2}\\ 2-a=0=\ \textgreater \ a=2\\ \frac{5-b}{\sqrt{2}+\sqrt{a} }=5\sqrt{2}=\ \textgreater \ b=-15\\ c\in R[/tex]