Question

当a不等于b时,根号下(x+a)(x+b)-根号下(x-a)(x-b)小趋于无穷的极限是

Answer 1

lim(x→inf.){sqrt[(x+a)(x+b)] - sqrt[(x-a)(x-b)]}

= lim(x→inf.)[(x+a)(x+b) - (x-a)(x-b)]/{sqrt[(x+a)(x+b)] + sqrt[(x-a)(x-b)]}

= 2(a+b)*lim(x→inf.)x/{sqrt[(x+a)(x+b)] + sqrt[(x-a)(x-b)]}

= 2(a+b)*lim(x→inf.)1/{sqrt[(1+a/x)(1+b/x)] + sqrt[(1-a/x)(1-b/x)]}

= a+b.