高数求极限问题，下图里这三道题解答过程都看不懂，麻烦高手讲解一下，最好详细点，谢谢啦！

Question

如题所述

Answer 1

(6)
lim(x->α) (sinx- sinα) /(x-α) (0/0)
=lim(x->α) cosx
=cosα

(7)
√(x^2+x) - √(x^2-x)
=[√(x^2+x) - √(x^2-x)] . [√(x^2+x) + √(x^2-x)]/[√(x^2+x) + √(x^2-x)]
= 2x/[√(x^2+x) + √(x^2-x)]

lim(x->∞) [√(x^2+x) - √(x^2-x) ]
=lim(x->∞) 2x/[√(x^2+x) + √(x^2-x)]
=lim(x->∞) 2/[√(1+1/x) + √(1-1/x)]
=2/(1+1)
=1

(8)
lim(x->0) [ ( 1- (1/2)x^2)^(2/3) -1 ]/[xln(1+x) ]
=lim(x->0) - (1/3)x^2 /x^2
=-1/3

x->0
( 1- (1/2)x^2)^(2/3) ~ 1 - (2/3)(1/2)x^2 = 1- (1/3)x^2
( 1- (1/2)x^2)^(2/3) -1 ~ - (1/3)x^2

ln(1+x) ~ x
xln(1+x) ~ x^2