Cho \(\mathop {\lim }\limits_{x \to 4} \frac{{\sqrt {3x + 4} - 4}}{{x - 4}} = \frac{a}{b}\) với \(\frac{a}{b}\) là phân số tối giản. Tính 2a + b².     

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