Cho hai dãy số (u_n) và (v_n).

a) Nếu \(\lim {u_n} = 2\) thì \(\lim \left( {{u_n} + 3} \right) = 6\).

b) Nếu \(\lim {u_n} = 2\) và \(\lim {v_n} = + \infty \) thì lim(u_n.v_n) = +∞.

c) Nếu u_n = 2n – 3 với n ∈ ℕ^* thì \(\lim \frac{{{u_n}}}{{n + 4}} = \frac{1}{2}\).

d) limu_n = −∞ với u_n = n³ – 5n + 6.

Tính $\underset{n\to +\infty }{\lim }\frac{3+3^2+3^3+\mathrm{...}+3^n}{4+4^2+4^3+\mathrm{...}+4^n}$

Cho các hàm số f(x), g(x), h(x) thỏa mãn \(\mathop {\lim }\limits_{x \to {x_0}} f\left( x \right) = 2;\mathop {\lim }\limits_{x \to {x_0}} g\left( x \right) = 5;\mathop {\lim }\limits_{x \to {x_0}} h\left( x \right) = 0\). Khi đó:

a) \(\mathop {\lim }\limits_{x \to {x_0}} \left[ {g\left( x \right) - f\left( x \right)} \right] = - 3\).

b) \(\mathop {\lim }\limits_{x \to {x_0}} \frac{{f\left( x \right)}}{{g\left( x \right)}} = \frac{2}{5}\).

c) \(\mathop {\lim }\limits_{x \to {x_0}} \frac{{h\left( x \right)}}{{g\left( x \right)}} = 0\).

d) \(\mathop {\lim }\limits_{x \to {x_0}} \left[ {g\left( x \right).h\left( x \right)} \right] = 0\).