Question

高数极限问题求详细过程解答！

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

Answer 1

1. 解：
原式 = { [ (n^2 + 17)^(1/2) – n ] * [
(n^2 + 17)^(1/2) + n ] } / { [ (n(n+1))^(1/2) - (n^2 - 1 )^(1/2)
] * [ (n(n+1))^(1/2) + (n^2 - 1 )^(1/2) ] } * { [
(n(n+1))^(1/2) + (n^2 - 1 )^(1/2) ] / [ (n^2 + 17)^(1/2) + n ] }
= { 17 } /
{ n+1 } * { [ (n(n+1))^(1/2) + (n^2
- 1 )^(1/2) ] / [ (n^2 + 17)^(1/2) + n ] }
= { (17/n )/ (1+1/n) } * { [ (1+1/n
))^(1/2) + (1 – 1/n^2 )^(1/2) ] / [ ( 1 + 17/n^2)^(1/2) + 1 ] } ==> { (0 )/ (1+0) } * { [ (1+0 ))^(1/2) + (1 – 0 )^(1/2) ] /
[ ( 1 + 0)^(1/2) + 1 ] }
= 0

2. 解：

/ e^(-1/x) * sin( 3/x^2 ) + x * arctan(1/x) / ≤ / e^(-1/x) / + (/x/) * (π/2)

ð 0 , ( xà 0- 时)，

故 原式 = 0

1. 解：

原式 =
[ 3- (1+x+x^2) ]/[(1-x) (1+x+x^2)]

= [(x+2)( 1-x ) ] / [(1-x) (1+x+x^2)]

= [(x+2)] / [(1+x+x^2)] è [(1+2)] / [(1+1+1^2)] = 2/3 , ( x-->1 时）