1.令 u = kx/e^x,当x->0时,u->0
lim(x->0) (1+kx/e^x)^(m/x) = lim(u->0) (1+u)^ [(1/u)*(mk/e^x)]= e^(mk)
lim(x->0) f(x) = lne^(mk) = mk
f(0) = mk
2.设函数 f(x) = x - a sinx - b,f(x) 在[0,a+b]上连续,且f(0)= - b,f(a+b)=a(1-sinx) ≥ 0
由闭区间上连续函数的性质--零值点定理,即得……
3.是的,dx/dy = 1/(dy/dx) = 1/ y'
4.lim(x->0) f(x) = lim(x->0) [(1+x)^(1/3)- (1-x)^(1/3)] / x
= lim(x->0) 【(1+x)^(1/3) - 1 + 1 - (1-x)^(1/3)】 / x 等价无穷小代换 (1+x)^(1/3) - 1 x/3
= lim(x->0) (x/3 + x/3) / x 1 - (1-x)^(1/3) x/3
= 2/3 ≠ f(0)
函数 f(x) 连续在 x = 0 不连续.