Proving a Limit Does Not Exist
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I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
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add a comment |
$begingroup$
I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
$endgroup$
add a comment |
$begingroup$
I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
$endgroup$
I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
calculus limits multivariable-calculus
edited 23 mins ago
Thomas Shelby
3,1771524
3,1771524
asked 36 mins ago
krauser126krauser126
394
394
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add a comment |
3 Answers
3
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oldest
votes
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HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
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add a comment |
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Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
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$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
$endgroup$
add a comment |
$begingroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
$endgroup$
add a comment |
$begingroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
$endgroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
answered 31 mins ago
Mark ViolaMark Viola
132k1275173
132k1275173
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$begingroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
$endgroup$
add a comment |
$begingroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
$endgroup$
add a comment |
$begingroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
$endgroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
answered 32 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
71729
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$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
$endgroup$
add a comment |
$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
$endgroup$
add a comment |
$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
$endgroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
answered 26 mins ago
Thomas ShelbyThomas Shelby
3,1771524
3,1771524
add a comment |
add a comment |
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