如果lim(x趋于0){根号下[1+f(x)(1-cos2x)]-1}/xsinx=6,则lim(x趋于0)f(x)=

Question

如果 lim(x趋于0) {根号下[1+f(x)(1-cos2x)] - 1}/xsinx = 6 ,则lim(x趋于0)f(x)=

Answer 1

显然x趋于0的时候,1-cos2x趋于0,
那么√[1+f(x)(1-cos2x)] - 1也趋于0,
而且等价于0.5* f(x)(1-cos2x),
而1-cos2x等价于0.5*(2x)^2即2x^2,
所以√[1+f(x)(1-cos2x)] - 1等价于 x^2 *f(x),
而x趋于0时,sinx也是x的等价无穷小,
故
lim(x趋于0) {√[1+f(x)(1-cos2x)] - 1} /xsinx
=lim(x趋于0) x^2 *f(x) /x^2
=lim(x趋于0) f(x)
=6