Question

极限问题。

极限问题。求详细解答。

Answer 1

以下过程供题主参考

追答

clear;
x = 1e-3
fun = @( t )( exp( x .* t ) - 1 ) ./ t
I = integral( fun , x / 2 , x )
test = I / sin(x^2)

Answer 2

lim(x->0) ∫ (x/2->x) [(e^(xt) -1 )/t ] dt / sin(x^2)
=lim(x->0) ∫ (x/2->x) [(e^(xt) -1 )/t ] dt / x^2 (0/0)
=lim(x->0) { [(e^(x^2) -1 )/x ] - [(e^(x^2/2) -1 )/x ] }/ (2x)
=lim(x->0) [ e^(x^2) - e^(x^2/2) ] / (2x^2) (0/0)
=lim(x->0) [ 2x.e^(x^2) - x.e^(x^2/2) ] / (4x)
=lim(x->0) [ 2(e^(x^2) - e^(x^2/2) ] / 4
= (2-1)/4
=1/4

Answer 3