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