epsilon-delta限制的定义据/h1>
已经有一个帐户?据A.href="//www.parkandroid.com/account/login/?next=/wiki/epsilon-delta-definition-of-a-limit/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">登录此处。据/a>
建议课程据/h4>
相关......据/h4>
- 微积分据/span>>据/span>
in.据A.href="//www.parkandroid.com/wiki/calculus/" class="wiki_link" title="微积分" target="_blank">微积分据/a>,这据strong> ε.据/span>-据span class="katex"> δ.据/span>一个定义据A.href="//www.parkandroid.com/wiki/limits-of-functions/" class="wiki_link" title="限制" target="_blank">限制据/a>评估A的极限是一种代数精确的制定据A.href="//www.parkandroid.com/wiki/functions/" class="wiki_link" title="功能" target="_blank">功能据/a>。非正式的定义指出限制据span class="katex"> L.据/span>一个函数在一个点据span class="katex"> X.据/span>0.据/span>如果无论如何都存在据span class="katex"> X.据/span>0.据/span>接近函数返回的值将始终接近据span class="katex"> L.据/span>。这一定义与用于评估在初等微积分的限制方法一致,但与此相关的数学严谨的语言出现在更高级别的分析。当据span class="katex"> ε.据/span>-据span class="katex"> δ.据/span>试图展示时定义也是有用据A.href="//www.parkandroid.com/wiki/continuous-functions/" class="wiki_link" title="函数的连续性。" target="_blank">函数的连续性。据/a>
在这篇文章中,我们将证明使用的Epsilon三角洲限制所有限制。据/p>
内容据/h4>
epsilon-delta限制的正式定义据/h2>
函数限制据span class="katex"> (据/span>ε.据/span>-据span class="katex"> δ.据/span>定义据span class="katex"> )据/span>
让我们据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>是在开放间隔内定义的函数据span class="katex"> X.据/span>0.据/span> (据/span>F.据/span>(据/span>X.据/span>0.据/span>)据/span>不需要定义据span class="katex"> )据/span>。据/span>我们说的限值据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>as.据span class="katex"> X.据/span>方法据span class="katex"> X.据/span>0.据/span>是据span class="katex"> L.据/span>,即据/p>
X.据/span>→据/span>X.据/span>0.据/span>林据/span>F.据/span>(据/span>X.据/span>)据/span>=据/span>L.据/span>那据/span>
如果是的话据span class="katex"> ε.据/span>>据/span>0.据/span>存在据span class="katex"> δ.据/span>>据/span>0.据/span>这一切据span class="katex"> X.据/span>
0.据/span>据据/span>|据/span>X.据/span>-据/span>X.据/span>0.据/span>|据/span>据据/span>δ.据/span>⟹据/span>|据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>L.据/span>|据/span>据据/span>ε.据/span>。据/span>
换句话说,定义指出,我们可以使函数返回的值据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>尽我们所希望的价值据span class="katex"> L.据/span>通过在一个足够小的间隔周围只用点据span class="katex"> X.据/span>0.据/span>。这个定义的一个有用的解释是可视化的两方,Alice和Bob之间的交换。首先,爱丽丝的挑战鲍勃,“我想,以确保这些值据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>将不是更远的据span class="katex"> ε.据/span>>据/span>0.据/span>从据span class="katex"> L.据/span>。“如果限制存在并且确实存在据span class="katex"> L.据/span>,然后鲍勃将能够通过给予她的价值来回应据span class="katex"> δ.据/span>,“如果为所有点据span class="katex"> X.据/span>在A之内据span class="katex"> δ.据/span>-radius间隔据span class="katex"> X.据/span>0.据/span>那么据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>将永远在一个据span class="katex"> ε.据/span>- interval据span class="katex"> L.据/span>”如果极限存在,那么Bob将能够向Alice的应对挑战,无论多么小的她选择据span class="katex"> ε.据/span>。据/span>
例如,在函数的图表中据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>下面,如果Alice给出了价值据span class="katex">
ε.据/span>,然后鲍勃给她数据span class="katex">
δ.据/span>这样的据span class="katex">
X.据/span>在开放间隔据span class="katex">
(据/span>X.据/span>0.据/span>-据/span>δ.据/span>那据/span>X.据/span>0.据/span>+据/span>δ.据/span>)据/span>,价值据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>位于间隔据span class="katex">
(据/span>L.据/span>-据/span>ε.据/span>那据/span>L.据/span>+据/span>ε.据/span>)据/span>。在这个例子中,就像爱丽丝一样据span class="katex">
ε.据/span>较小,更小,鲍勃总能找到一个较小的据span class="katex">
δ.据/span>满足此属性,显示限制存在。据/p>
随着Alice和Bob之间的交换表明,Alice通过提供价值来开始据span class="katex">
ε.据/span>然后知道这个值之后,Bob可以确定用于相应的值据span class="katex">
δ.据/span>。由于这个事件的顺序,值据span class="katex">
δ.据/span>通常给定为的函数据span class="katex">
ε.据/span>。请注意,可能有多个值据span class="katex">
δ.据/span>鲍勃可以给。据/p>
如果有任何价值据span class="katex">
ε.据/span>鲍勃找不到相应的鲍勃据span class="katex">
δ.据/span>,那么限制不存在!据/p>
对于这个功能据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>=据/span>3.据/span>X.据/span>2据/span>+据/span>2据/span>X.据/span>+据/span>1据/span>那据/span>爱丽丝想要鲍勃表明这一点据span class="katex">
X.据/span>→据/span>2据/span>林据/span>F.据/span>(据/span>X.据/span>)据/span>=据/span>1据/span>7.据/span>使用据span class="katex">
ε.据/span>-据span class="katex">
δ.据/span>限制的定义。据/p>
爱丽丝说,“我敢打赌,你不能选择一个实数据span class="katex">
δ.据/span>所以对于所有人来说据span class="katex">
X.据/span>in.据span class="katex">
(据/span>2据/span>-据/span>δ.据/span>那据/span>2据/span>+据/span>δ.据/span>)据/span>,我们有那个据span class="katex">
|据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>1据/span>7.据/span>|据/span>据据/span>0.据/span>。据/span>5.据/span>。“据/p>
以下哪一项选择最大据span class="katex">
δ.据/span>鲍勃可以给予他完成爱丽丝的挑战?据/p>
有限的无限限制据span class="katex"> X.据/span>
让我们据span class="katex"> F.据/span>被一个功能上包含数字一些开放间隔定义据span class="katex"> X.据/span>0.据/span> (据/span>F.据/span>(据/span>X.据/span>0.据/span>)据/span>不需要定义据span class="katex"> )据/span>。据/span>我们说的限值据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>as.据span class="katex"> X.据/span>方法据span class="katex"> X.据/span>0.据/span>偏向无限,即据span class="katex-display"> X.据/span>→据/span>X.据/span>0.据/span>林据/span>F.据/span>(据/span>X.据/span>)据/span>=据/span>∞据/span>如果每次正数据span class="katex"> m据/span>存在据span class="katex"> δ.据/span>>据/span>0.据/span>这一切据span class="katex"> X.据/span> 0.据/span>据据/span>|据/span>X.据/span>-据/span>X.据/span>0.据/span>|据/span>据据/span>δ.据/span>⟹据/span>F.据/span>(据/span>X.据/span>)据/span>>据/span>m据/span>
这是说的这些值据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>可以由通过拍摄任意大据span class="katex">
X.据/span>足够接近据span class="katex">
X.据/span>0.据/span>。据/p>
例如,在函数的图表中据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>。对于任何水平线据span class="katex">
y据/span>=据/span>m据/span>>据/span>0.据/span>,我们可以找到一个数字据span class="katex">
δ.据/span>>据/span>0.据/span>这样的任何间隔据span class="katex">
X.据/span>在间隔内据span class="katex">
(据/span>X.据/span>0.据/span>-据/span>δ.据/span>那据/span>X.据/span>0.据/span>+据/span>δ.据/span>)据/span>但是据span class="katex">
X.据/span>据/span>=据/span>X.据/span>0.据/span>,然后曲线据span class="katex">
y据/span>=据/span>F.据/span>(据/span>X.据/span>)据/span>谎言线以上据span class="katex">
y据/span>=据/span>m据/span>。我们可以看到任何更大的人据span class="katex">
m据/span>选择,较小的价值据span class="katex">
δ.据/span>是必要的。据/p>
在无限有限的限制据/h2>
在上一节中,我们使用了这些条款据E.m>任意大据/em>非正式地定义限制。即使他们对无限远处的限制描述良好,它们也不是在数学上严格的。让我们尝试正式定义它们。据/p>
F.据/span>据说有一个据E.m>限制在无穷远据/em>,如果存在实数据span class="katex"> L.据/span>这一切据span class="katex"> ε.据/span>>据/span>0.据/span>,存在据span class="katex"> N据/span>>据/span>0.据/span>这样据span class="katex"> |据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>L.据/span>|据/span>据据/span>ε.据/span>为所有人据span class="katex"> X.据/span>>据/span>N据/span>。换句话说,据span class="katex-display"> X.据/span>→据/span>∞据/span>林据/span>F.据/span>(据/span>X.据/span>)据/span>=据/span>L.据/span>。据/span>
例如,在函数的图表中据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>。对于足够大的价值据span class="katex">
N据/span>,我们可以找到间隔据span class="katex">
y据/span>是据span class="katex">
(据/span>L.据/span>-据/span>ε.据/span>那据/span>L.据/span>+据/span>ε.据/span>)据/span>但是据span class="katex">
y据/span>=据/span>L.据/span>,然后曲线据span class="katex">
y据/span>=据/span>F.据/span>(据/span>X.据/span>)据/span>在两条线之间据span class="katex">
y据/span>=据/span>L.据/span>-据/span>ε.据/span>和据span class="katex">
y据/span>=据/span>L.据/span>+据/span>ε.据/span>。我们可以看到任何更大的人据span class="katex">
N据/span>选择,较小的价值据span class="katex">
ε.据/span>是必要的。据/p>
表明据span class="katex">
X.据/span>→据/span>∞据/span>林据/span>(据/span>7.据/span>-据/span>X.据/span>1据/span>)据/span>=据/span>7.据/span>。据/span> 让我们据span class="katex">
ε.据/span>>据/span>0.据/span>那么据span class="katex">
N据/span>=据/span>ε.据/span>1据/span>。然后,对于所有据span class="katex">
X.据/span>>据/span>N据/span>那据span class="katex-display">
|据/span>|据/span>|据/span>|据/span>7.据/span>-据/span>X.据/span>1据/span>-据/span>7.据/span>|据/span>|据/span>|据/span>|据/span>=据/span>|据/span>|据/span>|据/span>|据/span>X.据/span>1据/span>|据/span>|据/span>|据/span>|据/span>=据/span>X.据/span>1据/span>据据/span>N据/span>1据/span>=据/span>ε.据/span>因此,我们的索赔。据span class="katex">
□据/span>
无限的无限限制据/h2>
F.据/span>据说有一个据E.m>在无限无量涨停据/em>,如果对所有据span class="katex"> m据/span>>据/span>0.据/span>,存在一个据span class="katex"> N据/span>>据/span>0.据/span>这样据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>>据/span>m据/span>为所有人据span class="katex"> X.据/span>>据/span>N据/span>。那是,据span class="katex"> X.据/span>→据/span>∞据/span>林据/span>F.据/span>(据/span>X.据/span>)据/span>=据/span>∞据/span>。据/p>
类似地,据span class="katex"> F.据/span>据说有一个据E.m>在无穷远处负无量涨停据/em>,如果对所有据span class="katex"> m据/span>据据/span>0.据/span>,存在一个据span class="katex"> N据/span>>据/span>0.据/span>这样据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>据据/span>m据/span>为所有人据span class="katex"> X.据/span>>据/span>N据/span>。那是,据span class="katex"> X.据/span>→据/span>∞据/span>林据/span>F.据/span>(据/span>X.据/span>)据/span>=据/span>-据/span>∞据/span>。据!-- end-definition -->
例如,在函数的图表中据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>。对于足够大的价值据span class="katex">
N据/span>,我们可以找到一个值据span class="katex">
m据/span>>据/span>0.据/span>这样据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>>据/span>m据/span>为所有人据span class="katex">
X.据/span>>据/span>N据/span>。据/p>
在无限内使用无限限制的正式定义,证明这一点据span class="katex">
X.据/span>→据/span>∞据/span>林据/span>X.据/span>9.据/span>=据/span>∞据/span>。据/p>
让我们据span class="katex">
m据/span>>据/span>0.据/span>,让我们据span class="katex">
N据/span>=据/span>9.据/span>m据/span>
。然后为所有人据span class="katex">
X.据/span>>据/span>N据/span>那据span class="katex-display">
X.据/span>9.据/span>>据/span>N据/span>9.据/span>=据/span>(据/span>9.据/span>m据/span>
)据/span>9.据/span>=据/span>m据/span>。据/span>□据/span> 从上面的例子中,我们知道,据span class="katex">
X.据/span>→据/span>∞据/span>林据/span>X.据/span>9.据/span>=据/span>∞据/span>。如果没有提取标量,则证明据span class="katex">
X.据/span>→据/span>∞据/span>林据/span>9.据/span>X.据/span>9.据/span>=据/span>∞据/span>使用Infinity的无限限制的正式定义。据/p>
让我们据span class="katex">
m据/span>>据/span>0.据/span>,让我们据span class="katex">
N据/span>=据/span>9.据/span>9.据/span>m据/span>
。然后为所有人据span class="katex">
X.据/span>>据/span>N据/span>那据span class="katex-display">
9.据/span>X.据/span>9.据/span>>据/span>9.据/span>N据/span>9.据/span>=据/span>9.据/span>(据/span>9.据/span>m据/span>
)据/span>9.据/span>=据/span>m据/span>。据/span>□据/span>
找到epsilon的三角洲据/h2>
在一般情况下,使用证明的限制据span class="katex">
ε.据/span>-据span class="katex">
δ.据/span>技术,我们必须找到一种表达据span class="katex">
δ.据/span>然后表明所需的不平等持有。表达式据span class="katex">
δ.据/span>最常是据span class="katex">
ε.据/span>那据/span>虽然有时它也是一个常数或更复杂的表达。以下是一些展示该属性的例子。据/p>
表明据span class="katex">
X.据/span>→据/span>π据/span>林据/span>X.据/span>=据/span>π据/span>。据/span> 让我们据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>=据/span>X.据/span>。据/span>首先,我们需要确定我们的价值据span class="katex">
δ.据/span>都将有。什么时候据span class="katex">
|据/span>X.据/span>-据/span>π据/span>|据/span>据据/span>δ.据/span>我们想要据span class="katex">
|据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>π据/span>|据/span>据据/span>ε.据/span>。据/span>我们知道据span class="katex">
|据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>π据/span>|据/span>=据/span>|据/span>X.据/span>-据/span>π据/span>|据/span>据据/span>δ.据/span>那据/span>所以服用据span class="katex">
δ.据/span>=据/span>ε.据/span>具有所需的性能。据/p>
还有其他价值据span class="katex">
δ.据/span>我们可以选择,如据span class="katex">
δ.据/span>=据/span>7.据/span>ε.据/span>。据/span>为什么这个值据span class="katex">
δ.据/span>也可接受?如果据span class="katex">
|据/span>X.据/span>-据/span>π据/span>|据/span>据据/span>δ.据/span>=据/span>7.据/span>ε.据/span>那据/span>然后据span class="katex">
|据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>π据/span>|据/span>据据/span>7.据/span>ε.据/span>据据/span>ε.据/span>那据/span>根据需要。据span class="katex">
□据/span> 要说明的是一个极限存在,我们并不一定需要证明的结果适用于所有据span class="katex">
ε.据/span>那据/span>但它足以证明结果适用于所有据span class="katex">
ε.据/span>据据/span>K.据/span>对于任何正价据span class="katex">
K.据/span>。据/span>这是因为据span class="katex">
δ.据/span>对应于特定值据span class="katex">
ε.据/span>=据/span>E.据/span>也是有效的据span class="katex">
δ.据/span>对于任何据span class="katex">
ε.据/span>>据/span>E.据/span>。据/span> 表明据/p>
X.据/span>→据/span>1据/span>林据/span>(据/span>5.据/span>X.据/span>-据/span>3.据/span>)据/span>=据/span>2据/span>。据/span> 在这个例子中,我们有据span class="katex">
X.据/span>0.据/span>=据/span>1据/span>那据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>=据/span>5.据/span>X.据/span>-据/span>3.据/span>,和据span class="katex">
L.据/span>=据/span>2据/span>从上面给出的限制的定义。对于任何一个据span class="katex">
ε.据/span>>据/span>0.据/span>由爱丽丝选择,鲍勃想找到据span class="katex">
δ.据/span>>据/span>0.据/span>这样的话据span class="katex">
X.据/span>在距离之内据span class="katex">
δ.据/span>of据span class="katex">
X.据/span>0.据/span>=据/span>1据/span>,即据/p>
|据/span>X.据/span>-据/span>1据/span>|据/span>据据/span>δ.据/span>那据/span> 然后据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>在距离之内据span class="katex">
ε.据/span>of据span class="katex">
L.据/span>=据/span>2据/span>,即据/p>
|据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>2据/span>|据/span>据据/span>ε.据/span>。据/span> 找到据span class="katex">
δ.据/span>,鲍勃向后落后据span class="katex">
ε.据/span>不等式:据/p>
|据/span>(据/span>5.据/span>X.据/span>-据/span>3.据/span>)据/span>-据/span>2据/span>|据/span>=据/span>|据/span>5.据/span>X.据/span>-据/span>5.据/span>|据/span>据据/span>ε.据/span>=据/span>5.据/span>|据/span>X.据/span>-据/span>1据/span>|据/span>据据/span>ε.据/span>=据/span>|据/span>X.据/span>-据/span>1据/span>|据/span>据据/span>5.据/span>ε.据/span>。据/span> 所以鲍勃给了爱丽丝的价值据span class="katex">
δ.据/span>=据/span>5.据/span>ε.据/span>。然后爱丽丝可以验证是否据span class="katex">
|据/span>X.据/span>-据/span>1据/span>|据/span>据据/span>δ.据/span>=据/span>5.据/span>ε.据/span>然后据/p>
|据/span>(据/span>5.据/span>X.据/span>-据/span>3.据/span>)据/span>-据/span>2据/span>|据/span>=据/span>|据/span>5.据/span>X.据/span>-据/span>5.据/span>|据/span>=据/span>5.据/span>|据/span>X.据/span>-据/span>1据/span>|据/span>据据/span>5.据/span>(据/span>5.据/span>ε.据/span>)据/span>=据/span>ε.据/span>。据/span>□据/span> 证明这一点据span class="katex">
X.据/span>→据/span>7.据/span>林据/span>(据/span>X.据/span>2据/span>+据/span>1据/span>)据/span>=据/span>5.据/span>0.据/span>。据/span> 让我们据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>=据/span>X.据/span>2据/span>+据/span>1据/span>。据/span>我们将首先确定我们的价值据span class="katex">
δ.据/span>应该是。什么时候据span class="katex">
|据/span>X.据/span>-据/span>7.据/span>|据/span>据据/span>δ.据/span>我们有据/p>
|据/span>X.据/span>2据/span>+据/span>1据/span>-据/span>5.据/span>0.据/span>|据/span>=据/span>|据/span>X.据/span>2据/span>-据/span>4.据/span>9.据/span>|据/span>=据/span>|据/span>X.据/span>-据/span>7.据/span>|据/span>|据/span>X.据/span>+据/span>7.据/span>|据/span>据据/span>δ.据/span>|据/span>X.据/span>+据/span>7.据/span>|据/span>。据/span> 假设据span class="katex">
|据/span>X.据/span>-据/span>7.据/span>|据/span>据据/span>1据/span>那据/span>我们有据span class="katex">
|据/span>X.据/span>|据/span>据据/span>8.据/span>那据/span>这意味着据span class="katex">
|据/span>X.据/span>+据/span>7.据/span>|据/span>据据/span>|据/span>X.据/span>|据/span>+据/span>|据/span>7.据/span>|据/span>=据/span>1据/span>5.据/span>由这件事据A.href="//www.parkandroid.com/wiki/triangle-inequality/" class="wiki_link" title="三角不等式" target="_blank">三角不等式据/a>。据/p>
所以,当我们让据span class="katex">
δ.据/span>=据/span>闵据/span>(据/span>1据/span>那据/span>1据/span>5.据/span>ε.据/span>)据/span>那据/span>我们将有据/p>
|据/span>X.据/span>2据/span>+据/span>1据/span>-据/span>5.据/span>0.据/span>|据/span>据据/span>δ.据/span>|据/span>X.据/span>+据/span>7.据/span>|据/span>据据/span>1据/span>5.据/span>δ.据/span>据据/span>ε.据/span>。据/span>□据/span> 使用据span class="katex">
ε.据/span>-据span class="katex">
δ.据/span>定义,证明以下限制:据/p>
X.据/span>→据/span>0.据/span>林据/span>X.据/span>罪恶据/span>X.据/span>=据/span>1据/span>。据/span> 在间隔上据span class="katex">
(据/span>-据/span>2据/span>π据/span>那据/span>2据/span>π据/span>)据/span>,我们有据span class="katex">
0.据/span>据据/span>COS.据/span>X.据/span>据据/span>X.据/span>罪恶据/span>X.据/span>据据/span>1据/span>,这意味着据/p>
|据/span>|据/span>|据/span>|据/span>X.据/span>罪恶据/span>X.据/span>-据/span>1据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>X.据/span>罪恶据/span>X.据/span>-据/span>1据/span>|据/span>|据/span>|据/span>|据/span>据据/span>1据/span>-据/span>COS.据/span>X.据/span>据据/span>2据/span>罪恶据/span>2据/span>2据/span>X.据/span>据据/span>2据/span>X.据/span>2据/span>。据/span> 现在,给予据span class="katex">
ε.据/span>>据/span>0.据/span>让据span class="katex">
δ.据/span>=据/span>ε.据/span>
。然后上面显示的计算即据span class="katex">
|据/span>X.据/span>-据/span>0.据/span>|据/span>据据/span>δ.据/span>=据/span>ε.据/span>
意思据/p>
|据/span>|据/span>|据/span>|据/span>X.据/span>罪恶据/span>X.据/span>-据/span>1据/span>|据/span>|据/span>|据/span>|据/span>据据/span>2据/span>X.据/span>2据/span>据据/span>2据/span>ε.据/span>
2据/span>据据/span>ε.据/span>那据/span> 完成证明。因此,据/p>
X.据/span>→据/span>0.据/span>林据/span>X.据/span>罪恶据/span>X.据/span>=据/span>1据/span>。据/span>□据/span> 有一个限制的情况据span class="katex">
F.据/span>(据/span>X.据/span>)据/span>as.据span class="katex">
X.据/span>方法据span class="katex">
X.据/span>0.据/span>没有绑定的增加或减少。在这种情况下,通常表示存在限制,并且该值是据A.href="//www.parkandroid.com/wiki/infinity/" class="wiki_link" title="无穷大" target="_blank">无穷大据/a>(或负无穷大)。然而,一些资源说,在这种情况下,限制不存在,仅仅因为这种限制使得微积分中的其他定理稍微更容易陈述并记住。据/p>
在证明限制时何时进入无限据span class="katex">
X.据/span>方法据span class="katex">
X.据/span>0.据/span>,这据span class="katex">
ε.据/span>-据span class="katex">
δ.据/span>不需要定义。相反,我们只需要表明该函数在靠近的值下任意大据span class="katex">
X.据/span>0.据/span>。据/span> 披据/p>
X.据/span>→据/span>∞据/span>林据/span>X.据/span>2据/span>1据/span>=据/span>0.据/span>。据/span> 给予据span class="katex">
ε.据/span>那据/span>我们需要选择据span class="katex">
δ.据/span>因此,如果据span class="katex">
X.据/span>>据/span>δ.据/span>那据/span>然后据span class="katex">
|据/span>|据/span>|据/span>|据/span>X.据/span>2据/span>1据/span>-据/span>0.据/span>|据/span>|据/span>|据/span>|据/span>据据/span>ε.据/span>。据/span> 假设据span class="katex">
|据/span>|据/span>X.据/span>2据/span>1据/span>-据/span>0.据/span>|据/span>|据/span>据据/span>ε.据/span>。据/span>在求解据span class="katex">
X.据/span>那据/span>我们发现,据span class="katex">
X.据/span>>据/span>ε.据/span>
1据/span>。据/span>因此,对于任何挑战据span class="katex">
ε.据/span>我们给出,确保了据span class="katex">
|据/span>|据/span>X.据/span>2据/span>1据/span>-据/span>0.据/span>|据/span>|据/span>据据/span>ε.据/span>只要我们选择据span class="katex">
δ.据/span>>据/span>ε.据/span>
1据/span>。据/span>因此据span class="katex">
林据/span>X.据/span>→据/span>∞据/span>X.据/span>2据/span>1据/span>=据/span>0.据/span>。据/span>□据/span>
披据/p>
X.据/span>→据/span>0.据/span>林据/span>X.据/span>2据/span>1据/span>=据/span>∞据/span>。据/span> 我们展示了任何正数据span class="katex">
L.据/span>,集据span class="katex">
δ.据/span>=据/span>L.据/span>
1据/span>。然后,当据span class="katex">
|据/span>X.据/span>-据/span>0.据/span>|据/span>据据/span>δ.据/span>,我们有据/p>
F.据/span>(据/span>X.据/span>)据/span>=据/span>X.据/span>2据/span>1据/span>>据/span>(据/span>L.据/span>
1据/span>)据/span>2据/span>1据/span>=据/span>L.据/span>。据/span> 这表明函数的值变得和住宿任意大据span class="katex">
X.据/span>接近零,或据span class="katex">
X.据/span>→据/span>0.据/span>林据/span>X.据/span>2据/span>1据/span>=据/span>∞据/span>。据/span>□据/span>
限制不存在据/h2>
当据span class="katex"> ε.据/span>-据span class="katex"> δ.据/span>定义可用于显示不存在点处的极限。对于存在的限制,我们的定义说:“每一个据span class="katex"> ε.据/span>>据/span>0.据/span>存在据span class="katex"> δ.据/span>>据/span>0.据/span>这样的话据span class="katex"> |据/span>X.据/span>-据/span>X.据/span>0.据/span>|据/span>据据/span>δ.据/span>那据/span>然后据span class="katex"> |据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>L.据/span>|据/span>据据/span>ε.据/span>。据/span>“这意味着据span class="katex"> L.据/span>是据strong>不是据/strong>限制如果存在据span class="katex"> ε.据/span>>据/span>0.据/span>这样别无选择据span class="katex"> δ.据/span>>据/span>0.据/span>确保据span class="katex"> |据/span>F.据/span>(据/span>X.据/span>)据/span>-据/span>L.据/span>|据/span>据据/span>ε.据/span>什么时候据span class="katex"> |据/span>X.据/span>-据/span>X.据/span>0.据/span>|据/span>据据/span>δ.据/span>。据/span>
考虑所提供的功能据/p>
F.据/span>(据/span>X.据/span>)据/span>=据/span>{据/span>1据/span>-据/span>1据/span>X.据/span>>据/span>0.据/span>X.据/span>据据/span>0.据/span>。据/span>
表明,在0极限不存在。据/p>
这几乎是显而易见的。我们看到,“右手极限”是据span class="katex"> 1据/span>那据/span>和“左侧限制”是据span class="katex"> -据/span>1据/span>。据/span>因此,它有意义限制不存在。让我们正式展示它,使用据span class="katex"> ε.据/span>-据span class="katex"> δ.据/span>我们上面开发的语言。据/p>
假设0处的限制存在并且等于据span class="katex"> L.据/span>。让我们据span class="katex"> ε.据/span>=据/span>2据/span>1据/span>,相应的据span class="katex"> δ.据/span>=据/span>δ.据/span>ε.据/span>>据/span>0.据/span>。据/p>
由于极限存在,我们知道,所有的据span class="katex"> y据/span>∈据/span>(据/span>-据/span>δ.据/span>那据/span>δ.据/span>)据/span>,我们有据span class="katex"> |据/span>F.据/span>(据/span>y据/span>)据/span>-据/span>L.据/span>|据/span>据据/span>ε.据/span>=据/span>2据/span>1据/span>。但是,我们也有据/p>
2据/span>=据/span>|据/span>|据/span>|据/span>|据/span>F.据/span>(据/span>2据/span>δ.据/span>)据/span>-据/span>F.据/span>(据/span>-据/span>2据/span>δ.据/span>)据/span>|据/span>|据/span>|据/span>|据/span>=据/span>|据/span>|据/span>|据/span>|据/span>F.据/span>(据/span>2据/span>δ.据/span>)据/span>-据/span>L.据/span>+据/span>L.据/span>-据/span>F.据/span>(据/span>-据/span>2据/span>δ.据/span>)据/span>|据/span>|据/span>|据/span>|据/span>≤.据/span>|据/span>|据/span>|据/span>|据/span>F.据/span>(据/span>2据/span>δ.据/span>)据/span>-据/span>L.据/span>|据/span>|据/span>|据/span>|据/span>+据/span>|据/span>|据/span>|据/span>|据/span>L.据/span>-据/span>F.据/span>(据/span>-据/span>2据/span>δ.据/span>)据/span>|据/span>|据/span>|据/span>|据/span>≤.据/span>2据/span>1据/span>+据/span>2据/span>1据/span>=据/span>1据/span>。据/span>
这是一个矛盾,所以我们的原始假设不是真的。据span class="katex"> □据/span>
上述证据很容易适应显示以下内容:据/p>
函数的域的内部点的限制仅存在且仅当左侧限制和右侧限制存在并且彼此相等时,才存在。据/p>
让我们据span class="katex"> F.据/span>(据/span>X.据/span>)据/span>是函数据span class="katex"> 0.据/span>什么时候据span class="katex"> X.据/span>是据A.href="//www.parkandroid.com/wiki/rational-numbers/" class="wiki_link" title="理性" target="_blank">理性据/a>和据span class="katex"> 1据/span>否则。表明据span class="katex"> X.据/span>→据/span>A.据/span>林据/span>F.据/span>(据/span>X.据/span>)据/span>不存在任何据span class="katex"> A.据/span>。据/span>
让我们据span class="katex"> A.据/span>是一个合理的数字。我们知道据span class="katex"> F.据/span>(据/span>A.据/span>)据/span>=据/span>0.据/span>。据/span>让我们据span class="katex"> ε.据/span>=据/span>0.据/span>。据/span>5.据/span>和据span class="katex"> δ.据/span>>据/span>0.据/span>。据/span>自从此以来据A.href="//www.parkandroid.com/wiki/irrational-numbers/" class="wiki_link" title="不合理的数字" target="_blank">不合理的数字据/a>在实数密集,我们可以找到一个不合理的数字据span class="katex"> B.据/span>∈据/span>(据/span>A.据/span>-据/span>δ.据/span>那据/span>A.据/span>+据/span>δ.据/span>)据/span>。据/span> |据/span>F.据/span>(据/span>B.据/span>)据/span>-据/span>F.据/span>(据/span>A.据/span>)据/span>|据/span>=据/span>1据/span>>据/span>ε.据/span>那据/span>但是据span class="katex"> |据/span>A.据/span>-据/span>B.据/span>|据/span>据据/span>δ.据/span>。据/span>因此,据span class="katex"> F.据/span>不连续据span class="katex"> A.据/span>。据/span>如果我们考虑的结果是相似的据span class="katex"> A.据/span>是一个不合理的点。据span class="katex"> □据/span>