a,
$\lim\dfrac{2^n-5^{n+1}}{1+5^n}$
$=\lim\dfrac{2^n-5^n.5}{1+5^n}$
$=\lim\dfrac{ \Big(\dfrac{2}{5}\Big)^n-5}{\dfrac{1}{5^n}+1}$
$=-5$
b,
$\lim\dfrac{n+2n+3n+...+n^2}{1+n^3}$
$=\lim\dfrac{n(1+2+3+...+n)}{1+n^3}$
$=\lim\dfrac{ \dfrac{n.n(n+1)}{2} }{n^3+1}$
$=\lim\dfrac{n^3+n^2}{2(n^3+1)}$
$=\lim\dfrac{1+\dfrac{1}{n}}{2+\dfrac{2}{n^3}}$
$=\dfrac{1}{2}$
c,
Tổng CSC $1+5+9+..+(4n+1)$ có $(4n+1-1):4+1= n+1$ số hạng.
$\lim\dfrac{ 1+5+9+...+(4n+1)}{2n^2+n+3}$
$=\lim\dfrac{ \dfrac{(4n+2)(n+1)}{2} }{2n^2+n+3}$
$=\lim\dfrac{(4n+2)(n+1) }{2(2n^2+n+3)}$
$=\dfrac{\Big(4+\dfrac{2}{n}\Big).\Big(1+\dfrac{1}{n}\Big) }{4+\dfrac{2}{n}+\dfrac{6}{n^2}}$
$=1$
