第3章 3.10

(1)

$\lim_{x\to 0}\frac{5x^3-8x^2}{3x^4-16x^2}=\lim_{x\to 0}\frac{5x-8}{3x^2-16}=\lim_{x\to 0}\frac{-8}{-16}=\frac{1}{2}$

(2)

$\lim_{x\to 0}\frac{3x^3-2x^2}{x^2-x}=\lim_{x\to 0}\frac{3x^2-2x}{x-1}=0$

(3)

$\lim_{x\to 0}\frac{1}{x}\left\{1-\frac{1}{(1+x)^3}\right\}=\lim_{x\to 0}\frac{1}{x}\left\{\frac{1+3x+3x^2+x^3-1}{(1+x)^3}\right\}=\frac{1}{x}\left\{\frac{3x+3x^2+x^3}{(1+x)^3}\right\}=\frac{3+3x+x^2}{(1+x)^3}=3$

(4)

$\lim_{x\to 0}\frac{1}{x}\left\{\frac{1}{(x-2)^2}-\frac{1}{4}\right\}=\lim_{x\to 0}\frac{1}{x}\left\{\frac{4-x^2+4x-4}{4(x-2)^2}\right\}=\lim_{x\to 0}\frac{-x+4}{4(x-2)^2}=\frac{1}{4}$