Mixing
Functions in $mathbb R^2$ and double limits in Compositions
Clash Royale CLAN TAG#URR8PPP
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I was looking at the double limits for two variable function in $mathbb R$, and so I decided to partition the set of all such functions into two disjoint subsets as follows:
$$mathbb F=f(x,y):(x,y) in mathbb R^2=mathbb F_i cup mathbb F_j quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad,,(0)$$
$$mathbb F_i(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) neq lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad(1)$$
$$mathbb F_j(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) = lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad,(2)$$
Here are some examples of subsets of (1) and (2) for $X,Y=0,infty$:
$$ left frac 1xy,xy,rm e^xy,
rm e^-xy,ln left( xy right) ,tan left( xy right)
right subset mathbb F_i quadquadquadquadquadquadquadquadquad,quadquadquadquadquadquadquadquadquad(1e)$$
$$ left x^y,frac xy,frac yx,cos left( frac y
x right) ,sin left( frac yx right) ,tan left( frac x
y right) ,tan left( frac yx right) right
subset mathbb F_jquadquadquadquadquadquadquadquadquad,quadquadquad(2e)$$
And in observing the obvious trend that can be seen in these examples for many other functions, I made the following statements:
$$S_0:lim _xrightarrow X left( lim _yrightarrow Yg left( x,y
right) right) =lim _yrightarrow Y left( lim _xrightarrow X
g left( x,y right) right)$$
$$S_1: X=lim _yrightarrow YBigr(frac1YBigl), lor , X=lim _xrightarrow XBigr(frac1YBigl) $$
$$S_2:lim _xrightarrow X left( lim _yrightarrow Y(f(g left( x,y
)right)) right) =lim _yrightarrow Y left( lim _xrightarrow X
f(g left( x,y right)) right) $$
$$S_0 land S_1 Rightarrow S_2$$
So my first question is:
1) Are the above assertions always the case, and what functions serve as counter examples?
And my second question is:
2) Does there exist any such case examples for which $mathbb F_i neq ,$ is true but $S_1$ is false?
real-analysis limits functions
asked Jul 24 at 17:15
Adam
11411
add a comment |Â
up vote
0
down vote
favorite
I was looking at the double limits for two variable function in $mathbb R$, and so I decided to partition the set of all such functions into two disjoint subsets as follows:
$$mathbb F=f(x,y):(x,y) in mathbb R^2=mathbb F_i cup mathbb F_j quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad,,(0)$$
$$mathbb F_i(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) neq lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad(1)$$
$$mathbb F_j(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) = lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad,(2)$$
Here are some examples of subsets of (1) and (2) for $X,Y=0,infty$:
$$ left frac 1xy,xy,rm e^xy,
rm e^-xy,ln left( xy right) ,tan left( xy right)
right subset mathbb F_i quadquadquadquadquadquadquadquadquad,quadquadquadquadquadquadquadquadquad(1e)$$
$$ left x^y,frac xy,frac yx,cos left( frac y
x right) ,sin left( frac yx right) ,tan left( frac x
y right) ,tan left( frac yx right) right
subset mathbb F_jquadquadquadquadquadquadquadquadquad,quadquadquad(2e)$$
And in observing the obvious trend that can be seen in these examples for many other functions, I made the following statements:
$$S_0:lim _xrightarrow X left( lim _yrightarrow Yg left( x,y
right) right) =lim _yrightarrow Y left( lim _xrightarrow X
g left( x,y right) right)$$
$$S_1: X=lim _yrightarrow YBigr(frac1YBigl), lor , X=lim _xrightarrow XBigr(frac1YBigl) $$
$$S_2:lim _xrightarrow X left( lim _yrightarrow Y(f(g left( x,y
)right)) right) =lim _yrightarrow Y left( lim _xrightarrow X
f(g left( x,y right)) right) $$
$$S_0 land S_1 Rightarrow S_2$$
So my first question is:
1) Are the above assertions always the case, and what functions serve as counter examples?
And my second question is:
2) Does there exist any such case examples for which $mathbb F_i neq ,$ is true but $S_1$ is false?
real-analysis limits functions
asked Jul 24 at 17:15
Adam
11411
Why use $i,j$ as subscripts?
– zhw.
Jul 24 at 19:32
what should I use?
– Adam
Jul 24 at 19:50
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I was looking at the double limits for two variable function in $mathbb R$, and so I decided to partition the set of all such functions into two disjoint subsets as follows:
$$mathbb F=f(x,y):(x,y) in mathbb R^2=mathbb F_i cup mathbb F_j quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad,,(0)$$
$$mathbb F_i(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) neq lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad(1)$$
$$mathbb F_j(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) = lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad,(2)$$
Here are some examples of subsets of (1) and (2) for $X,Y=0,infty$:
$$ left frac 1xy,xy,rm e^xy,
rm e^-xy,ln left( xy right) ,tan left( xy right)
right subset mathbb F_i quadquadquadquadquadquadquadquadquad,quadquadquadquadquadquadquadquadquad(1e)$$
$$ left x^y,frac xy,frac yx,cos left( frac y
x right) ,sin left( frac yx right) ,tan left( frac x
y right) ,tan left( frac yx right) right
subset mathbb F_jquadquadquadquadquadquadquadquadquad,quadquadquad(2e)$$
And in observing the obvious trend that can be seen in these examples for many other functions, I made the following statements:
$$S_0:lim _xrightarrow X left( lim _yrightarrow Yg left( x,y
right) right) =lim _yrightarrow Y left( lim _xrightarrow X
g left( x,y right) right)$$
$$S_1: X=lim _yrightarrow YBigr(frac1YBigl), lor , X=lim _xrightarrow XBigr(frac1YBigl) $$
$$S_2:lim _xrightarrow X left( lim _yrightarrow Y(f(g left( x,y
)right)) right) =lim _yrightarrow Y left( lim _xrightarrow X
f(g left( x,y right)) right) $$
$$S_0 land S_1 Rightarrow S_2$$
So my first question is:
1) Are the above assertions always the case, and what functions serve as counter examples?
And my second question is:
2) Does there exist any such case examples for which $mathbb F_i neq ,$ is true but $S_1$ is false?
real-analysis limits functions
asked Jul 24 at 17:15
Adam
11411
I was looking at the double limits for two variable function in $mathbb R$, and so I decided to partition the set of all such functions into two disjoint subsets as follows:
$$mathbb F=f(x,y):(x,y) in mathbb R^2=mathbb F_i cup mathbb F_j quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad,,(0)$$
$$mathbb F_i(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) neq lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad(1)$$
$$mathbb F_j(f(X,Y))=Bigglf(x,y) in mathbb F:lim _xrightarrow X left( lim _yrightarrow Yf left( x,y
right) right) = lim _yrightarrow Y left( lim _x
rightarrow X f left( x,y right) right)Biggr
quadquadquadquadquadquadquad,(2)$$
Here are some examples of subsets of (1) and (2) for $X,Y=0,infty$:
$$ left frac 1xy,xy,rm e^xy,
rm e^-xy,ln left( xy right) ,tan left( xy right)
right subset mathbb F_i quadquadquadquadquadquadquadquadquad,quadquadquadquadquadquadquadquadquad(1e)$$
$$ left x^y,frac xy,frac yx,cos left( frac y
x right) ,sin left( frac yx right) ,tan left( frac x
y right) ,tan left( frac yx right) right
subset mathbb F_jquadquadquadquadquadquadquadquadquad,quadquadquad(2e)$$
And in observing the obvious trend that can be seen in these examples for many other functions, I made the following statements:
$$S_0:lim _xrightarrow X left( lim _yrightarrow Yg left( x,y
right) right) =lim _yrightarrow Y left( lim _xrightarrow X
g left( x,y right) right)$$
$$S_1: X=lim _yrightarrow YBigr(frac1YBigl), lor , X=lim _xrightarrow XBigr(frac1YBigl) $$
$$S_2:lim _xrightarrow X left( lim _yrightarrow Y(f(g left( x,y
)right)) right) =lim _yrightarrow Y left( lim _xrightarrow X
f(g left( x,y right)) right) $$
$$S_0 land S_1 Rightarrow S_2$$
So my first question is:
1) Are the above assertions always the case, and what functions serve as counter examples?
And my second question is:
2) Does there exist any such case examples for which $mathbb F_i neq ,$ is true but $S_1$ is false?
real-analysis limits functions
asked Jul 24 at 17:15
Adam
11411
edited Aug 2 at 11:57
asked Jul 24 at 17:15
Adam
11411
asked Jul 24 at 17:15
Adam
11411
asked Jul 24 at 17:15
Adam
11411
11411
Why use $i,j$ as subscripts?
– zhw.
Jul 24 at 19:32
what should I use?
– Adam
Jul 24 at 19:50
add a comment |Â
Why use $i,j$ as subscripts?
– zhw.
Jul 24 at 19:32
what should I use?
– Adam
Jul 24 at 19:50
Why use $i,j$ as subscripts?
– zhw.
Jul 24 at 19:32
Why use $i,j$ as subscripts?
– zhw.
Jul 24 at 19:32
what should I use?
– Adam
Jul 24 at 19:50
what should I use?
– Adam
Jul 24 at 19:50
add a comment |Â
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Why use $i,j$ as subscripts?
– zhw.
Jul 24 at 19:32
what should I use?
– Adam
Jul 24 at 19:50