Question

求下列曲线的渐近线

求下列曲线的渐近线:(1) y=x^2/(x^2+2x-3) (2) y=(x^2-3x+2)/(1-x^2)

Answer 1

y=x²/(x²+2x-3)
lim f(x)
=lim x²/(x²+2x-3)
=lim f(x)1/(1+2/x-3/x²)
=1
y=x²/(x²+2x-3)的水平渐近线为y=1
lim f(x)
=lim x²/(x²+2x-3)
当x²+2x-3=0即x0=-3或1时,lim f(x)=∞
y=x²/(x²+2x-3)的铅直渐近线为x=-3和x=1
lim f(x)/x
=lim x²/x(x²+2x-3)
=lim x/(x²+2x-3)
=lim (1/x)/(1+2/x-3/x²)
=0
y=x²/(x²+2x-3)没有斜渐近线
y=(x²-3x+2)/(1-x²)
lim f(x)
=lim (x²-3x+2)/(1-x²)
=lim (1-3/x+2/x²)/(1/x²-1)
=-1
y=(x²-3x+2)/(1-x²)的水平渐近线为y=-1
lim f(x)
=lim (x²-3x+2)/(1-x²)
当1-x²=0即x0=-1或1时,lim f(x)=∞
y=(x²-3x+2)/(1-x²)的铅直渐近线为x=-1和x=1
lim f(x)/x
=lim (x²-3x+2)/x(1-x²)
=lim (1/x-3/x²+2/x³)/(1/x²-1)
=0
y=(x²-3x+2)/(1-x²)没有斜渐近线
若lim f(x)=a,则y=a为f(x)的水平渐近线
若lim f(x)=∞,则x=x0为f(x)的铅直渐近线
若lim f(x)/x=k、lim [f(x)-kx]=b,则y=kx+b为f(x)的斜渐近线