极限定义的问题。

Question

ForAll[\[CurlyEpsilon], \[CurlyEpsilon] > 0, 
Exists[\[Delta], \[Delta] > 0, 
Implies[0 < Abs[x] < \[Delta], 
0 < Abs[f[x]] < \[CurlyEpsilon]]]] // Resolve
ForAll[\[CurlyEpsilon], \[CurlyEpsilon] > 0, 
Exists[\[Delta], \[Delta] > 0, 
Implies[0 < RealAbs[x] < \[Delta], 
0 < RealAbs[f[x]] < \[CurlyEpsilon]]]] // Resolve

True

\!\(
\*SubscriptBox[\(\[ForAll]\), \({\[CurlyEpsilon]}, \[CurlyEpsilon] > 
   0\)]\(
\*SubscriptBox[\(\[Exists]\), \({\[Delta]}, \[Delta] > 
    0\)]\((0 < RealAbs[x] < \[Delta] \[Implies] 
    0 < RealAbs[f[x]] < \[CurlyEpsilon])\)\)\)

 

为什么Abs换成RealAbs就不行了？

Comments

你这是哪个版本，10.0的没有RealAbs这个函数

---

mathematica11.1

Answer 1

In[5]:= ForAll[\[CurlyEpsilon], \[CurlyEpsilon] > 0, 
  Exists[\[Delta], \[Delta] > 0, 
   Implies[0 < RealAbs[x] < \[Delta], 
    0 < Abs[f[x]] < \[CurlyEpsilon]]]] // Resolve
ForAll[\[CurlyEpsilon], \[CurlyEpsilon] > 0, 
  Exists[\[Delta], \[Delta] > 0, 
   Implies[0 < RealAbs[x] < \[Delta], 
    0 < RealAbs[f[x]] < \[CurlyEpsilon]]]] // Resolve

True

\!\(
\*SubscriptBox[\(\[ForAll]\), \({\[CurlyEpsilon]}, \[CurlyEpsilon] > 
   0\)]\(
\*SubscriptBox[\(\[Exists]\), \({\[Delta]}, \[Delta] > 
    0\)]\((0 < RealAbs[x] < \[Delta] \[Implies] 
    0 < RealAbs[f[x]] < \[CurlyEpsilon])\)\)\)

Out[5]= \!\(
\*SubscriptBox[\(\[ForAll]\), \({\[CurlyEpsilon]}, \[CurlyEpsilon] > 
   0\)]\(
\*SubscriptBox[\(\[Exists]\), \({\[Delta]}, \[Delta] > 
    0\)]\((0 < RealAbs[x] < \[Delta] \[Implies] 
    0 < Abs[f[x]] < \[CurlyEpsilon])\)\)\)

Out[6]= \!\(
\*SubscriptBox[\(\[ForAll]\), \({\[CurlyEpsilon]}, \[CurlyEpsilon] > 
   0\)]\(
\*SubscriptBox[\(\[Exists]\), \({\[Delta]}, \[Delta] > 
    0\)]\((0 < RealAbs[x] < \[Delta] \[Implies] 
    0 < RealAbs[f[x]] < \[CurlyEpsilon])\)\)\)

Out[7]= True

Out[8]= \!\(
\*SubscriptBox[\(\[ForAll]\), \({\[CurlyEpsilon]}, \[CurlyEpsilon] > 
   0\)]\(
\*SubscriptBox[\(\[Exists]\), \({\[Delta]}, \[Delta] > 
    0\)]\((0 < RealAbs[x] < \[Delta] \[Implies] 
    0 < RealAbs[f[x]] < \[CurlyEpsilon])\)\)\)

还是不太明白您的意思