第5章 5.10

(1)

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

(2)

$\lim_{x \to \infty} \frac{2x-1}{x^2-x+1} = \lim_{x \to \infty} \frac{\dfrac{2}{x}-\dfrac{1}{x^2}}{1-\dfrac{1}{x}+\dfrac{1}{x^2}} = 0$

(3)

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

(4)

$\lim_{x \to \infty} \left(\sqrt{x^2+4x+1}-x\right) = \lim_{x \to \infty} \frac{x^2+4x+1-x^2}{\sqrt{x^2+4x+1}+x} = \lim_{x \to \infty} \frac{4x+1}{\sqrt{x^2+4x+1}+x} = \lim_{x \to \infty} \frac{4+\dfrac{1}{x}}{\sqrt{1+\dfrac{4}{x}+\dfrac{1}{x^2}}+1} = 2$

(5)

$\lim_{x \to \infty} \left(\sqrt{x^2+2x}-x\right) = \lim_{x \to \infty} \frac{x^2+2x-x^2}{\sqrt{x^2+2x}+x} = \lim_{x \to \infty} \frac{2x}{\sqrt{x^2+2x}+x} = \lim_{x \to \infty} \frac{2}{\sqrt{1+\dfrac{2}{x}}+1} = 1$

(6)

$\lim_{x \to -\infty} \left(\sqrt{x^2+2x}-x\right) = \lim_{x \to \infty} \left(\sqrt{x^2-2x}+x\right) = \lim_{x \to \infty} \frac{x^2-2x-x^2}{\sqrt{x^2-2x}-x} = \lim_{x \to \infty} \frac{-2x}{\sqrt{x^2-2x}-x} = \lim_{x \to \infty} \frac{-2}{\sqrt{1-\dfrac{2}{x}}-1} = \infty$