Question

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)

请写出详细过程 谢谢!

Answer 1

y=x²/(x²+2x-3)

lim<x→∞>f(x)
=lim<x→∞>x²/(x²+2x-3)
=lim<x→∞>f(x)1/(1+2/x-3/x²)
=1

y=x²/(x²+2x-3)的水平渐近线为y=1

lim<x→x0>f(x)
=lim<x→x0>x²/(x²+2x-3)
当x²+2x-3=0即x0=-3或1时，lim<x→x0>f(x)=∞

y=x²/(x²+2x-3)的铅直渐近线为x=-3和x=1

lim<x→∞>f(x)/x
=lim<x→∞>x²/x(x²+2x-3)
=lim<x→∞>x/(x²+2x-3)
=lim<x→∞>(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<x→∞>f(x)
=lim<x→∞>(x²-3x+2)/(1-x²)
=lim<x→∞>(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<x→x0>f(x)
=lim<x→x0>(x²-3x+2)/(1-x²)
当1-x²=0即x0=-1或1时，lim<x→x0>f(x)=∞

y=(x²-3x+2)/(1-x²)的铅直渐近线为x=-1和x=1

lim<x→∞>f(x)/x
=lim<x→∞>(x²-3x+2)/x(1-x²)
=lim<x→∞>(1/x-3/x²+2/x³)/(1/x²-1)
=0

y=(x²-3x+2)/(1-x²)没有斜渐近线

若lim<x→∞>f(x)=a，则y=a为f(x)的水平渐近线
若lim<x→x0>f(x)=∞，则x=x0为f(x)的铅直渐近线
若lim<x→∞>f(x)/x=k、lim<x→∞>[f(x)-kx]=b，则y=kx+b为f(x)的斜渐近线