Question

设F(x)=∫(0趋向x) [(x-t)f(t)dt]/(sinx)^2 求lim(x趋向0)F(x),f(0)存在,怎么做

Answer 1

本题少了个条件，f(x)连续
lim(x趋向0)F(x)
=lim(x趋向0)∫(0--->x)
[(x-t)f(t)dt]/(sinx)²
分母用等价无穷小代换
=lim(x趋向0)∫(0--->x)
[(x-t)f(t)dt]/x²
分子拆开
=lim(x趋向0)
[x∫(0--->x)
f(t)dt
-
∫(0--->x)
tf(t)dt
]/x²
洛必达法则
=lim(x趋向0)
[∫(0--->x)
f(t)dt+xf(x)
-
xf(x)
]/(2x)
=lim(x趋向0)
∫(0--->x)
f(t)dt
/(2x)
洛必达法则
=lim(x趋向0)
f(x)
/2
=f(0)/2