用定义证明下列极限:lim x趋向于π/4 sinx=二分之根号二
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  • 求证:lim(x->π/4) sinx = √2/2 = sin(π/4)

    证明:

    ① 对任意 ε>0 ,

    ∵ √2/2 = sin(π/4) ,|cosx| ≤ 1 ,|sinx|≤|x|

    ∴要使 | sinx - √2/2| < ε 成立,

    即只要满足:|sinx - √2/2| = | sinx - sin(π/4)| = |2cos[(x+π/4)/2]*sin[(x-π/4)/2]|

    ≤ |2sin[(x-π/4)/2]| ≤|2[(x-π/4)/2]| =|(x-π/4)|< ε 即可.

    ② 故存在 δ = ε > 0

    ③ 当 | x-π/4 |< δ =ε 时,

    ④ 恒有:|sinx - √2/2 | < ε 成立.

    ∴ lim(x->π/4) sinx = √2/2

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