Question

描绘下列函数图形y=2+3x/(x+1)^2

Answer 1

y=2+(3x+3-3)/(x+1)^2=2+3/(x+1)-3/(x+1)^2,x不等于-1

y`=-3/(x+1)^2+6/(x+1)^3,令y`=0,解得x=1

y``=6/(x+1)^3-18/(x+1)^4,令y``=0,解得x=2

x(-∝,-1)-1(-1,1)1(1,2)2(2,+∝)

y`-+0+-

y``---0+

现在求渐近线

水平渐近线y=lim(x→+∝)2+3x/(x+1)^2=2

铅直渐近线x=lim(x→-1)2+3x/(x+1)^2 =-∝,

倾斜渐近线斜率k=lim(x→+∝)[2+3x/(x+1)^2]/x=0