Question

函数f(x)=(x^2-1)|x|/(x-1)(x-2)在R上有几条渐近线?

Answer 1

f(x)=(x^2-1)|x|/[(x-1)(x-2)],
limf(x)=∞,f(x)无水平渐近线.
limf(x)=lim(x+1)/(x-2)=-2,x=1 不是垂直渐近线.
limf(x)=∞,则 x=2 是垂直渐近线.
k=lim[f(x)/x]=lim[(x+1)/(x-2)]=1,
b=lim[f(x)-x]=lim[3x/(x-2)]=3,y=x+3 是斜渐近线.
k=lim[f(x)/x]=lim[-(x+1)/(x-2)]=-1,
b=lim[f(x)-x]=lim[-3x/(x-2)]=-3,y=-x-3 是斜渐近线.
共有3条渐近线.