Tính giới hạn Lim {{x + {x^2} + ... + {x^{2024}} - 2024/ x + {x^2} + {x^3} + {x^4} + {x^5} - 5

+ Ta có \(\mathop {\lim }\limits_{x \to 1} \frac{{\left( {x - 1} \right) + \left( {{x^2} - 1} \right) + ... + \left( {{x^{2024}} - 1} \right)}}{{\left( {x - 1} \right) + \left( {{x^2} - 1} \right) + \left( {{x^3} - 1} \right) + \left( {{x^4} - 1} \right) + \left( {{x^5} - 1} \right)}}\)

=\(\mathop {\lim }\limits_{x \to 1} \frac{{\left( {x - 1} \right)\left( {1 + \left( {x + 1} \right) + \left( {{x^2} + x + 1} \right) + ... + \left( {{x^{2023}} + {x^{2022}} + ... + x + 1} \right)} \right)}}{{\left( {x - 1} \right)\left( {1 + \left( {x + 1} \right) + ... + \left( {{x^4} + {x^3} + {x^2} + x + 1} \right)} \right)}}\)

 

=\(\mathop {\lim }\limits_{x \to 1} \frac{{1 + \left( {x + 1} \right) + \left( {{x^2} + x + 1} \right) + ... + \left( {{x^{2023}} + {x^{2022}} + ... + x + 1} \right)}}{{1 + \left( {x + 1} \right) + ... + \left( {{x^4} + {x^3} + {x^2} + x + 1} \right)}}\) =\(\mathop {\lim }\limits_{x \to 1} \frac{{1 + 2 + 3 + ... + 2024}}{{1 + 2 + ... + 5}}\)=136620