Bài 5.1 trang 109

Đề bài

Tìm các giới hạn sau:

a) $\underset{n\to +\infty }{lim}\frac{n^2+n+1}{2n^2+1}$;            

b) $\underset{n\to +\infty }{lim}\left(\sqrt{n^2+2n}-n\right)$.

Lời giải chi tiết

a) $\underset{n\to +\infty }{lim}\frac{n^2+n+1}{2n^2+1}$

$=\underset{n\to +\infty }{lim}\frac{1+\frac{1}{n}+\frac{1}{n^2}}{2+\frac{1}{n^2}}=\frac{1}{2}$.

b) $\underset{n\to +\infty }{lim}(\sqrt{n^2+2n}-n)$

$=\underset{n\to +\infty }{lim}\frac{2n}{\sqrt{n^2+2n}+n}$

$=\underset{n\to +\infty }{lim}\frac{2}{\sqrt{1+\frac{2}{n}}+1}=1$.