(4)
∫ [1/(2√x) +e^(-x) - sin5x) ] dx
分开积分
=∫ dx/(2√x) +∫e^(-x) dx - ∫sin5x dx
利用 d(-x)=-dx , d(5x)=5dx
=∫ dx/(2√x) -∫e^(-x) d(-x) -(1/5) ∫sin5x d5x
=√x -e^(-x) +(1/5)cos5x +C
得出结果
∫ [1/(2√x) +e^(-x) - sin5x) ] dx =√x -e^(-x) +(1/5)cos5x +C
(5)
lim(x->0, y->0) sin(x^2.y)/(x^2+y^2)
=lim(x->0, y->0) (x^2.y)/(x^2+y^2)
y=kx
lim(x->0) (x^2.(kx)/(x^2+(kx)^2)
=lim(x->0) kx/(1+k^2)
=0
(3)
不一定
e.g
f(x) =|x|
x=0, f'(x) 不存在
但
lim(x->0) f(x) =0