数学极限lim(x→0)((x^3)/(3*x^2-1)) lim(x→+无穷)(√x(x+2)-√(x^2-x+1)) lim(x→1)(tan(x-1))/(x^2-1)

题目

数学极限lim(x→0)((x^3)/(3*x^2-1)) lim(x→+无穷)(√x(x+2)-√(x^2-x+1)) lim(x→1)(tan(x-1))/(x^2-1)

答案

lim(x→0)((x^3)/(3*x^2-1))
=lim(x→0)3x^2/6x=0
lim(x→+∞)(√x(x+2)-√(x^2-x+1)) =lim(x→+∞)(√(x+1)^2-1-√x-1/2)^2+3/4)
= lim(x→+∞)(x+1-(x-1/2))
=3/2
lim(x→1)(tan(x-1))/(x^2-1)
=lim(x→1)(sin(x-1)/cos(x-1)(x^2-1)
=lim(x→1)(sin(x-1)/cos(x-1)(x-1)(x+1) 当x-1→0 limsinx=limx
所以lim(x→1)(sin(x-1)/cos(x-1)(x-1)(x+1)
=lim(x→1)1/cos(x-1)(x+1)
=1/2