\(\lim\dfrac{2n^2+n}{n^2+4}=\lim\dfrac{n^2\left(2+\dfrac{1}{n}\right)}{n^2\left(1+\dfrac{4}{n^2}\right)}=\lim\dfrac{2+\dfrac{1}{n}}{1+\dfrac{4}{n^2}}=\dfrac{2+0}{1+0}=2\)
\(\lim\dfrac{6n+2}{n+5}=\lim\dfrac{n\left(6+\dfrac{2}{n}\right)}{n\left(1+\dfrac{5}{n}\right)}=\lim\dfrac{6+\dfrac{2}{n}}{1+\dfrac{5}{n}}=\dfrac{6+0}{1+0}=6\)
\(\lim\dfrac{2.3^n+5^n}{5^n+3^n}=\lim\dfrac{5^n\left[2.\left(\dfrac{3}{5}\right)^n+1\right]}{5^n\left[1+\left(\dfrac{3}{5}\right)^n\right]}=\lim\dfrac{2.\left(\dfrac{3}{5}\right)^n+1}{1+\left(\dfrac{3}{5}\right)^n}=\dfrac{2.0+1}{1+0}=1\)
a) \(\lim\limits_{n\rightarrow\infty}\dfrac{2n^2+n}{n^2+4}=\)\(\lim\limits_{n\rightarrow\infty}\dfrac{n^2\left(2+\dfrac{1}{n}\right)}{n^2\left(1+\dfrac{4}{n^2}\right)}=\)\(\lim\limits_{n\rightarrow\infty}\dfrac{2+\dfrac{1}{n}}{1+\dfrac{4}{n^2}}=2\left(đpcm\right)\)
