Question

tanx /tan3x 当 x趋向于 派/2时的极限

Answer 1

无穷/无穷
洛必达
上下同求导
lim=(tanx)' /(tan3x)'
=[1/(1+x^2)]/[3/(1+(3x)^2)]
=[1+9x^2]/[3(1+x^2)]
上下同除x^2
=[9+1/x^2]/[3+3/x^2]->(9+0)/(3+0)=3
所以极限为3

Answer 2

原式=tanx(1-tanxtan2x)/(tanx+tan2x)
=tanx*[1-tanx*2tanx/(1-tan²x)]/(tanx+2tanx/(1-tan²x)]
=(1-3tan²x)/(3-tan²x)
=(cos²x-3sin²x)/(3cos²x-sin²x)
所以lim tanx /tan3x
=(0-3*1)/(3*0-1)
=-3

Answer 3

1 三角变形 lim {4(cosx)^2-3} / {3-4(sinx)^2} =-3 / (-1)=3
2 等价替换=lim {sinx sin(3x-3π/2)} / {-sin(x- π/2) sin3x}
=lim {sin(3x-3π/2)} / {sin(x- π/2)}
= lim {3x-3π/2} / {x- π/2}
=3