第5章 5.1

(1)

$\lim_{x \to \infty} \frac{4x-1}{2x-3} = \lim_{x \to \infty} \frac{4 - \dfrac{1}{x}}{2 - \dfrac{3}{x}} = 2$

(2)

$\lim_{x \to \infty} \frac{2x^2}{3x^2+2} = \lim_{x \to \infty} \frac{2}{3 + \dfrac{2}{x^2}} = \frac{2}{3}$

(3)

$\lim_{x \to \infty} \frac{2x^2+3x+1}{x^2+2x+2} = \lim_{x \to \infty} \frac{2 + \dfrac{3}{x} + \dfrac{1}{x^2}}{1 + \dfrac{2}{x} + \dfrac{2}{x^2}} = 2$

(4)

$\lim_{x \to -\infty} \frac{3x^3-2x^2-1}{2x^3-4x+5} = \lim_{x \to -\infty} \frac{3 - \dfrac{2}{x} - \dfrac{1}{x^3}}{2 - \dfrac{4}{x^2} + \dfrac{5}{x^3}} = \frac{3}{2}$

(5)

$\lim_{x \to \infty} \frac{2x}{3x^2+4} = \lim_{x \to \infty} \frac{\dfrac{2}{x}}{3 + \dfrac{4}{x^2}} = 0$

(6)

$\lim_{x \to -\infty} \frac{2x^3}{x^2-x+1} = \lim_{x \to -\infty} \frac{2x}{1 - \dfrac{1}{x} + \dfrac{1}{x^2}} = -\infty$