a) \(\lim \frac{{8{n^2} + n}}{{{n^2}}} = \lim \left( {8 + \frac{1}{n}} \right) = \lim 8 + \lim \frac{1}{n} = 8 + 0 = 8\)                     

b) \(\lim \frac{{\sqrt {4 + {n^2}} }}{n} = \lim \frac{{n\sqrt {\frac{4}{{{n^2}}} + 1} }}{n} = \sqrt {\lim \left( {\frac{4}{{{n^2}}} + 1} \right)}  = \sqrt {0 + 1}  = 1\)

a) \(\lim \frac{{5n + 1}}{{2n}} = \lim \frac{{5 + \frac{1}{n}}}{2} = \frac{{5 + 0}}{2} = \frac{5}{2}\)           

b) \(\lim \frac{{6{n^2} + 8n + 1}}{{5{n^2} + 3}} = \lim \frac{{6 + \frac{8}{n} + \frac{1}{{{n^2}}}}}{{5 + \frac{3}{{{n^2}}}}} = \frac{{6 + 0 + 0}}{{5 + 0}} = \frac{6}{5}\)                   

c) \(\lim \frac{{\sqrt {{n^2} + 5n + 3} }}{{6n + 2}} = \lim \frac{{\sqrt {1 + \frac{5}{n} + \frac{3}{{{n^2}}}} }}{{6 + \frac{2}{n}}} = \frac{{\sqrt {1 + 0 + 0} }}{{6 + 0}} = \frac{1}{6}\)

d) \(\lim \left( {2 - \frac{1}{{{3^n}}}} \right) = \lim 2 - \lim {\left( {\frac{1}{3}} \right)^n} = 2 - 0 = 0\)              

e) \(\lim \frac{{{3^n} + {2^n}}}{{{{4.3}^n}}} = \lim \frac{{1 + {{\left( {\frac{2}{3}} \right)}^n}}}{4} = \frac{{1 + 0}}{4} = \frac{1}{4}\)                       

g) \(\lim \frac{{2 + \frac{1}{n}}}{{{3^n}}}\)

Ta có \(\lim \left( {2 + \frac{1}{n}} \right) = \lim 2 + \lim \frac{1}{n} = 2 + 0 = 2 > 0;\lim {3^n} =  + \infty  \Rightarrow \lim \frac{{2 + \frac{1}{n}}}{{{3^n}}} = 0\)

a) \(\lim \frac{{3n - 1}}{n} = \lim \frac{{n\left( {3 - \frac{1}{n}} \right)}}{n} = \lim \left( {3 - \frac{1}{n}} \right) = 3 - 0 = 3\)

b) \(\lim \frac{{\sqrt {{n^2} + 2} }}{n} = \lim \frac{{\sqrt {{n^2}\left( {1 + \frac{2}{{{n^2}}}} \right)} }}{n} = \lim \frac{{n\sqrt {1 + \frac{2}{{{n^2}}}} }}{n} = \lim \sqrt {1 + \frac{2}{{{n^2}}}}  = 1 + 0 = 1\)

c) \(\lim \frac{2}{{3n + 1}} = \lim \frac{2}{{n\left( {3 + \frac{1}{n}} \right)}} = \lim \left( {\frac{2}{n}.\frac{1}{{3 + \frac{1}{n}}}} \right) = \lim \frac{2}{n}.\lim \frac{1}{{3 + \frac{1}{n}}} = 0.\frac{1}{{3 + 0}} = 0\)

d) \(\lim \frac{{\left( {n + 1} \right)\left( {2n + 2} \right)}}{{{n^2}}} = \lim \frac{{n\left( {1 + \frac{1}{n}} \right).2n\left( {1 + \frac{1}{n}} \right)}}{{{n^2}}} = \lim \frac{{2{n^2}{{\left( {1 + \frac{1}{n}} \right)}^2}}}{{{n^2}}}\)

                                                      \( = \lim 2{\left( {1 + \frac{1}{n}} \right)^2} = 2.{\left( {1 + 0} \right)^2} = 2\)

a) \(\lim \frac{{2{n^2} + 6n + 1}}{{8{n^2} + 5}} = \lim \frac{{{n^2}\left( {2 + \frac{6}{n} + \frac{1}{{{n^2}}}} \right)}}{{{n^2}\left( {8 + \frac{5}{{{n^2}}}} \right)}} = \lim \frac{{2 + \frac{6}{n} + \frac{1}{n}}}{{8 + \frac{5}{n}}} = \frac{2}{8} = \frac{1}{4}\)

b) \(\lim \frac{{4{n^2} - 3n + 1}}{{ - 3{n^3} + 6{n^2} - 2}} = \lim \frac{{{n^3}\left( {\frac{4}{n} - \frac{3}{{{n^2}}} + \frac{1}{{{n^3}}}} \right)}}{{{n^3}\left( { - 3 + \frac{6}{n} - \frac{2}{{{n^3}}}} \right)}} = \lim \frac{{\frac{4}{n} - \frac{3}{{{n^2}}} + \frac{1}{{{n^3}}}}}{{ - 3 + \frac{6}{n} - \frac{2}{{{n^3}}}}} = \frac{{0 - 0 + 0}}{{ - 3 + 0 - 0}} = 0\).

c) \(\lim \frac{{\sqrt {4{n^2} - n + 3} }}{{8n - 5}} = \lim \frac{{n\sqrt {4 - \frac{1}{n} + \frac{3}{{{n^2}}}} }}{{n\left( {8 - \frac{5}{n}} \right)}} = \frac{{\sqrt {4 - 0 + 0} }}{{8 - 0}} = \frac{2}{8} = \frac{1}{4}\).

d) \(\lim \left( {4 - \frac{{{2^{{\rm{n}} + 1}}}}{{{3^{\rm{n}}}}}} \right) = \lim \left( {4 - 2 \cdot {{\left( {\frac{2}{3}} \right)}^{\rm{n}}}} \right) = 4 - 2.0 = 4\).

e) \(\lim \frac{{{{4.5}^{\rm{n}}} + {2^{{\rm{n}} + 2}}}}{{{{6.5}^{\rm{n}}}}} = \lim \frac{{{{4.5}^{\rm{n}}} + {2^2}{{.2}^{\rm{n}}}}}{{{{6.5}^{\rm{n}}}}} = \lim \frac{{{5^n}.\left[ {4 + 4.{{\left( {\frac{2}{5}} \right)}^{\rm{n}}}} \right]}}{{{{6.5}^n}}} = \lim \frac{{4 + 4.{{\left( {\frac{2}{5}} \right)}^{\rm{n}}}}}{6} = \frac{{4 + 4.0}}{6} = \frac{2}{3}\).

g) \(\lim \frac{{2 + \frac{4}{{{n^3}}}}}{{{6^{\rm{n}}}}} = \lim \left( {2 + \frac{4}{{{{\rm{n}}^3}}}} \right).\lim {\left( {\frac{1}{6}} \right)^{\rm{n}}} = \left( {2 + 0} \right).0 = 0\).

a) \(\lim\limits\dfrac{2n^2+3n}{n^2+1}=\lim\limits\dfrac{n^2\left(2+\dfrac{3n}{n^2}\right)}{n^2\left(1+\dfrac{1}{n^2}\right)}=\lim\limits\dfrac{2+\dfrac{3}{n}}{1+\dfrac{1}{n^2}}=2\).

b) \(\lim\limits\dfrac{\sqrt{4n^2+3}}{n}\\ =\lim\limits\dfrac{\sqrt{n^2\left(4+\dfrac{3}{n^2}\right)}}{n}\\ =\lim\limits\dfrac{\sqrt[n]{4+\dfrac{3}{n^2}}}{n}\\ =\lim\limits\sqrt{4+\dfrac{3}{n^2}}\\ =2.\)

a) \(\lim \frac{{ - 2n + 1}}{n} = \lim \frac{{n\left( { - 2 + \frac{1}{n}} \right)}}{n} = \lim \left( { - 2 + \frac{1}{n}} \right) =  - 2\)

b) \(\lim \frac{{\sqrt {16{n^2} - 2} }}{n} = \lim \frac{{\sqrt {{n^2}\left( {16 - \frac{2}{{{n^2}}}} \right)} }}{n} = \lim \frac{{n\sqrt {16 - \frac{2}{{{n^2}}}} }}{n} = \lim \sqrt {16 - \frac{2}{{{n^2}}}}  = 4\)

c) \(\lim \frac{4}{{2n + 1}} = \lim \frac{4}{{n\left( {2 + \frac{1}{n}} \right)}} = \lim \left( {\frac{4}{n}.\frac{1}{{2 + \frac{1}{n}}}} \right) = \lim \frac{4}{n}.\lim \frac{1}{{2 + \frac{1}{n}}} = 0\)

d) \(\lim \frac{{{n^2} - 2n + 3}}{{2{n^2}}} = \lim \frac{{{n^2}\left( {1 - \frac{2}{n} + \frac{3}{{{n^2}}}} \right)}}{{2{n^2}}} = \lim \frac{{1 - \frac{2}{n} + \frac{3}{{{n^2}}}}}{2} = \frac{1}{2}\)

a) Đặt \({u_n} = 2 + {\left( {\frac{2}{3}} \right)^n} \Leftrightarrow {u_n} - 2 = {\left( {\frac{2}{3}} \right)^n}\).

Suy ra \(\lim \left( {{u_n} - 2} \right) = \lim {\left( {\frac{2}{3}} \right)^n} = 0\)

Theo định nghĩa, ta có \(\lim {u_n} = 2\). Vậy \(\lim \left( {2 + {{\left( {\frac{2}{3}} \right)}^n}} \right) = 2\)

b) Đặt \({u_n} = \frac{{1 - 4n}}{n} = \frac{1}{n} - 4 \Leftrightarrow {u_n} - \left( { - 4} \right) = \frac{1}{n}\).

Suy ra \(\lim \left( {{u_n} - \left( { - 4} \right)} \right) = \lim \frac{1}{n} = 0\).

Theo định nghĩa, ta có \(\lim {u_n} =  - 4\). Vậy \(\lim \left( {\frac{{1 - 4n}}{n}} \right) =  - 4\)

a) \(\begin{array}{l}\lim {u_n} = \lim \left( {3 + \frac{1}{n}} \right) = \lim 3 + \lim \frac{1}{n} = 3 + 0 = 3\\\lim {v_n} = \lim \left( {5 - \frac{2}{{{n^2}}}} \right) = \lim 5 - \lim \frac{2}{{{n^2}}} = 5 - 0 = 5\end{array}\)

b)

\(\begin{array}{l}\lim \left( {{u_n} + {v_n}} \right) = \lim {u_n} + \lim {v_n} = 3 + 5 = 8\\\lim \left( {{u_n} - {v_n}} \right) = \lim {u_n} - \lim {v_n} = 3 - 5 =  - 2\\\lim \left( {{u_n}.{v_n}} \right) = \lim {u_n}.\lim {v_n} = 3.5 = 15\\\lim \frac{{{u_n}}}{{{v_n}}} = \frac{{\lim {u_n}}}{{\lim {v_n}}} = \frac{3}{5}\end{array}\)

a) Đặt \(f\left( x \right) = 2{x^2} - x\).

Hàm số \(y = f\left( x \right)\) xác định trên \(\mathbb{R}\).

Giả sử \(\left( {{x_n}} \right)\) là dãy số bất kì thỏa mãn \({x_n} \to 3\) khi \(n \to  + \infty \). Ta có:

\(\lim f\left( {{x_n}} \right) = \lim \left( {2x_n^2 - {x_n}} \right) = 2.\lim x_n^2 - \lim {x_n} = {2.3^2} - 3 = 15\).

Vậy \(\mathop {\lim }\limits_{x \to 3} \left( {2{x^2} - x} \right) = 15\).

b) Đặt \(f\left( x \right) = \frac{{{x^2} + 2x + 1}}{{x + 1}}\).

Hàm số \(y = f\left( x \right)\) xác định trên \(\mathbb{R}\).

Giả sử \(\left( {{x_n}} \right)\) là dãy số bất kì thỏa mãn \({x_n} \to  - 1\) khi \(n \to  + \infty \). Ta có:

\(\lim f\left( {{x_n}} \right) = \lim \frac{{x_n^2 + 2{x_n} + 1}}{{{x_n} + 1}} = \lim \frac{{{{\left( {{x_n} + 1} \right)}^2}}}{{{x_n} + 1}} = \lim \left( {{x_n} + 1} \right) = \lim {x_n} + 1 =  - 1 + 1 = 0\).

Vậy \(\mathop {\lim }\limits_{x \to  - 1} \frac{{{x^2} + 2x + 1}}{{x + 1}} = 0\).