y=cosx/x 0＜x≤1 是有界函数吗？ 函数y=xcosx在内是否有界
\u51fd\u6570y=1/x\u662f\u6709\u754c\u51fd\u6570\u5417\uff1f\u82e5\u5b58\u5728\u6b63\u6570M\uff0c\u4f7f\u5f97\u5bf9\u51fd\u6570f(x)\u5b9a\u4e49\u57df\u5185\u7684\u4efb\u610fx\u90fd\u6709|f(x)|\u2264M\uff0c\u5219\u79f0\u51fd\u6570f(x)\u4e3a\u6709\u754c\u51fd\u6570\u3002\u6709\u65f6\u6211\u4eec\u4e5f\u4f1a\u628a\u4e0d\u7b49\u5f0f|f(x)|\u2264M\u5199\u6210-M\u2264f(x)\u2264M\uff0c\u56e0\u6b64\u51fd\u6570\u6709\u754c\u5c31\u662f\u6307\u51fd\u6570\u65e2\u6709\u4e0b\u754c\u53c8\u6709\u4e0a\u754c\u3002
\u4ece\u51fd\u6570\u56fe\u50cf\u4e0a\u770b\uff0c\u6709\u754c\u51fd\u6570\u7684\u56fe\u5f62\u7279\u5f81\u5c31\u662f\u4e0d\u8bbax\u53d6\u5b9a\u4e49\u57df\u5185\u7684\u4f55\u503cf(x)\u7684\u56fe\u50cf\u59cb\u7ec8\u5728\u5728\u4e24\u6761\u5e73\u884c\u7ebfy=-M\u4e0ey=M\u4e4b\u95f4\u3002
\u7531\u4e8ex>1>0\uff0c\u6240\u4ee50<1/x<1\uff0c\u53730<f(x)=1/x<1\uff0c\u6545y=1/x\u57281\u5230\u6b63\u65e0\u7a77\u4e0a\u662f\u6709\u754c\u51fd\u6570\u3002
\u5e0c\u671b\u80fd\u89e3\u51b3\u60a8\u7684\u95ee\u9898\u3002
\u65e0\u754c\u3002
y= x cosx
\u53d6x(n) = 2n\u03c0, \u5f53n->+\u221e\u65f6, y(n) = x(n) cos(x(n)) = 2n\u03c0 ->+\u221e
\u6545 x cosx \u5728(-\u221e,+\u221e)\u4e0a\u65e0\u754c\u3002
x\u2192\uff0b\u221elimy\u2018=-xsinx+cosx \u2208R
\u56fe\u8c61\u4e3a\u5728y=x\u4e0ey=-x\u95f4\u4ee52\u03c0\u4e3a\u5468\u671f\u6765\u56de\u632f\u8361\uff0c\u5373-\u258fx\u258f\u2266y\u2266\u258fx\u258f\uff0c
\u56e0\u4e3ay=x\u4e0ey=-x\u5411\u6b63\u8d1f\u65b9\u5411\u7684\u6781\u9650\u90fd\u4e0d\u6536\u655b\uff0c\u6240\u4ee5y=xcosx\u5728\uff08\uff0d\u221e\uff0c\uff0b\u221e\uff09\u5185\u65e0\u754c\u3002
\u65e2\u65e0\u4e0a\u754c\u4e5f\u65e0\u4e0b\u754c\u3002
\u6269\u5c55\u8d44\u6599\u6709\u754c\u51fd\u6570
\u5b9a\u4e491\u8bbef\u5b9a\u4e49\u5728D\u4e0a\uff0c
\u82e5∃M\u2208R\uff0c∀x\u2208D\uff0cf(x)<=M\uff0c\u5219\u79f0f\u5728D\u4e0a\u6709\u4e0a\u754c\uff1b
\u82e5∃L\u2208R\uff0c∀x\u2208D\uff0cf(x)>=L\uff0c\u5219\u79f0f\u5728D\u4e0a\u6709\u4e0b\u754c\u3002
\u6709\u4e0a\u4e0b\u754c\u7684\u7b49\u4ef7\u6761\u4ef6\uff1a
f\u5728D\u4e0a\u6709\u4e0a\u754c ∃M>0\uff0c∀x\u2208D\uff0cf(x)<=M
f\u5728D\u4e0a\u6709\u4e0a\u754c ∃L>0\uff0c∀x\u2208D\uff0cf(x)>=L
lim【x→0+】y=lim【x→0+】(cosx/x)=∞
y(1)=(cos1)/1=常数
所以：y是单边有界函数。
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