求函数y=(x-1)e^(π\2+arctanx)的水平、铅直、斜渐近线.

Question

求函数y=(x-1)e^(π\2+arctanx)的水平、铅直、斜渐近线.

Answer 1

y=(x-1)e^(π/2+arctanx)
lim(x→0)(x-1)e^(π/2+arctanx)=-e^(π/2)
水平渐近线y=-e^(π/2)
(x-1)e^arctanx=ye^(-π/2)
lim(y→0)ye^(-π/2)=0
lim(x→1)(x-1)e^(arctanx)=0
垂直渐近线x=1
lim(x→+∞)(x-1)e^(π/2+arctanx)=(x-1)e^π
x>0斜渐近线y=(x-1)e^π (x>0)
lim(x→-∞)(x-1)e^(π/2+arctanx)=(x-1)
x