Question

Tìm các giới hạn sau:

a) \(\lim \left( {2 + {{\left( {\frac{2}{3}} \right)}^n}} \right)\);         

b) \(\lim \left( {\frac{{1 - 4n}}{n}} \right)\).

Answer 1

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 \(\lim \left( {\frac{{1 - 4n}}{n}} \right) =  - 4\)