Use the $epsilon - delta$ definition to verify that $lim_{(x,y)to(0,0)}frac{x^3-y^3}{x^2+y^2} = 0$?...
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Finding the limit of a 2-dimensional function
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I know that I have to find a $epsilon > 0$ s.t. |$frac{x^3-y^3}{x^2+y^2}$|<$epsilon$ whenever $0<sqrt{x^2+y^2}<delta$ but what are the steps to proving this?
I'm looking for an answer which shows the steps as well as explains the intuition.
Thanks
calculus limits multivariable-calculus
marked as duplicate by Decaf-Math, StubbornAtom, Trevor Gunn, GNUSupporter 8964民主女神 地下教會, amWhy
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Nov 21 at 19:40
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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up vote
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favorite
This question already has an answer here:
Finding the limit of a 2-dimensional function
2 answers
I know that I have to find a $epsilon > 0$ s.t. |$frac{x^3-y^3}{x^2+y^2}$|<$epsilon$ whenever $0<sqrt{x^2+y^2}<delta$ but what are the steps to proving this?
I'm looking for an answer which shows the steps as well as explains the intuition.
Thanks
calculus limits multivariable-calculus
marked as duplicate by Decaf-Math, StubbornAtom, Trevor Gunn, GNUSupporter 8964民主女神 地下教會, amWhy
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Nov 21 at 19:40
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This one is asking for it "by definition" whereas the other answers use shortcuts.
– DanielV
Nov 21 at 18:20
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up vote
-3
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favorite
up vote
-3
down vote
favorite
This question already has an answer here:
Finding the limit of a 2-dimensional function
2 answers
I know that I have to find a $epsilon > 0$ s.t. |$frac{x^3-y^3}{x^2+y^2}$|<$epsilon$ whenever $0<sqrt{x^2+y^2}<delta$ but what are the steps to proving this?
I'm looking for an answer which shows the steps as well as explains the intuition.
Thanks
calculus limits multivariable-calculus
This question already has an answer here:
Finding the limit of a 2-dimensional function
2 answers
I know that I have to find a $epsilon > 0$ s.t. |$frac{x^3-y^3}{x^2+y^2}$|<$epsilon$ whenever $0<sqrt{x^2+y^2}<delta$ but what are the steps to proving this?
I'm looking for an answer which shows the steps as well as explains the intuition.
Thanks
This question already has an answer here:
Finding the limit of a 2-dimensional function
2 answers
calculus limits multivariable-calculus
calculus limits multivariable-calculus
edited Nov 21 at 17:37
StubbornAtom
5,00411138
5,00411138
asked Nov 21 at 17:29
K.M.
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marked as duplicate by Decaf-Math, StubbornAtom, Trevor Gunn, GNUSupporter 8964民主女神 地下教會, amWhy
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Nov 21 at 19:40
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Nov 21 at 19:40
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
This one is asking for it "by definition" whereas the other answers use shortcuts.
– DanielV
Nov 21 at 18:20
add a comment |
1
This one is asking for it "by definition" whereas the other answers use shortcuts.
– DanielV
Nov 21 at 18:20
1
1
This one is asking for it "by definition" whereas the other answers use shortcuts.
– DanielV
Nov 21 at 18:20
This one is asking for it "by definition" whereas the other answers use shortcuts.
– DanielV
Nov 21 at 18:20
add a comment |
2 Answers
2
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1
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accepted
Numerator:
$|(x^3-y^3)|=$
$|(x-y)|(x^2+xy+y^2)|le$
$|x-y||x^2+y^2| +|xy|| le$
$|x-y||x^2+y^2| +|x|^2+|y|^2) le$
$2|x-y||x^2+y^2| le$
$ 2(|x|+|y|) (x^2+y^2)le$
$4 sqrt{x^2+y^2}(x^2+y^2)$.
Finally
$dfrac{|x^3-y^3|}{x^2+y^2} le 4sqrt{x^2+y^2}.$
Choose $delta =epsilon/4.$
Used: $x^2+y^2 ge 2|xy|;$
$|x| =sqrt{x^2} le sqrt{x^2+y^2}$,
$|y| =sqrt{y^2} le sqrt{x^2+y^2}$.
add a comment |
up vote
0
down vote
$$(forall y > 0 ~ exists r > 0 ~ forall bar x )~ 0 < | bar x - bar c | < r implies |f bar x - L| < y$$
$(forall y > 0 ~ exists r > 0 ~ forall bar x )$ is equivalent to asking for the existence of a function $g$ that, given a maximum deviation from $L$, outputs the neighborhood $r$ that satisfies that maximum deviation. In other words, find the disc centered at $bar c$ with radius $g(y)$ such that every value of $f$ within that region is within $y$ of $L$. Symbolically:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x - bar c | < g(y) implies |f bar x - L| < y$$
Putting in the values for your problem:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x | < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
or to put more tersely, find $g : mathbb R to mathbb R$ such that for $bar x ne (0, 0)$
$$|bar x| < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
A polar substitution helps:
$$r < g(y) implies left|frac{(rcos t)^3 - (rsin t)^3}{(rcos t)^2 + (rsin t)^2}right| < y$$
Can you find the $g$ now? One more hint if you need it:
$-1 le cos t, sin t le 1$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Numerator:
$|(x^3-y^3)|=$
$|(x-y)|(x^2+xy+y^2)|le$
$|x-y||x^2+y^2| +|xy|| le$
$|x-y||x^2+y^2| +|x|^2+|y|^2) le$
$2|x-y||x^2+y^2| le$
$ 2(|x|+|y|) (x^2+y^2)le$
$4 sqrt{x^2+y^2}(x^2+y^2)$.
Finally
$dfrac{|x^3-y^3|}{x^2+y^2} le 4sqrt{x^2+y^2}.$
Choose $delta =epsilon/4.$
Used: $x^2+y^2 ge 2|xy|;$
$|x| =sqrt{x^2} le sqrt{x^2+y^2}$,
$|y| =sqrt{y^2} le sqrt{x^2+y^2}$.
add a comment |
up vote
1
down vote
accepted
Numerator:
$|(x^3-y^3)|=$
$|(x-y)|(x^2+xy+y^2)|le$
$|x-y||x^2+y^2| +|xy|| le$
$|x-y||x^2+y^2| +|x|^2+|y|^2) le$
$2|x-y||x^2+y^2| le$
$ 2(|x|+|y|) (x^2+y^2)le$
$4 sqrt{x^2+y^2}(x^2+y^2)$.
Finally
$dfrac{|x^3-y^3|}{x^2+y^2} le 4sqrt{x^2+y^2}.$
Choose $delta =epsilon/4.$
Used: $x^2+y^2 ge 2|xy|;$
$|x| =sqrt{x^2} le sqrt{x^2+y^2}$,
$|y| =sqrt{y^2} le sqrt{x^2+y^2}$.
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Numerator:
$|(x^3-y^3)|=$
$|(x-y)|(x^2+xy+y^2)|le$
$|x-y||x^2+y^2| +|xy|| le$
$|x-y||x^2+y^2| +|x|^2+|y|^2) le$
$2|x-y||x^2+y^2| le$
$ 2(|x|+|y|) (x^2+y^2)le$
$4 sqrt{x^2+y^2}(x^2+y^2)$.
Finally
$dfrac{|x^3-y^3|}{x^2+y^2} le 4sqrt{x^2+y^2}.$
Choose $delta =epsilon/4.$
Used: $x^2+y^2 ge 2|xy|;$
$|x| =sqrt{x^2} le sqrt{x^2+y^2}$,
$|y| =sqrt{y^2} le sqrt{x^2+y^2}$.
Numerator:
$|(x^3-y^3)|=$
$|(x-y)|(x^2+xy+y^2)|le$
$|x-y||x^2+y^2| +|xy|| le$
$|x-y||x^2+y^2| +|x|^2+|y|^2) le$
$2|x-y||x^2+y^2| le$
$ 2(|x|+|y|) (x^2+y^2)le$
$4 sqrt{x^2+y^2}(x^2+y^2)$.
Finally
$dfrac{|x^3-y^3|}{x^2+y^2} le 4sqrt{x^2+y^2}.$
Choose $delta =epsilon/4.$
Used: $x^2+y^2 ge 2|xy|;$
$|x| =sqrt{x^2} le sqrt{x^2+y^2}$,
$|y| =sqrt{y^2} le sqrt{x^2+y^2}$.
edited Nov 21 at 19:10
answered Nov 21 at 19:05
Peter Szilas
10.5k2720
10.5k2720
add a comment |
add a comment |
up vote
0
down vote
$$(forall y > 0 ~ exists r > 0 ~ forall bar x )~ 0 < | bar x - bar c | < r implies |f bar x - L| < y$$
$(forall y > 0 ~ exists r > 0 ~ forall bar x )$ is equivalent to asking for the existence of a function $g$ that, given a maximum deviation from $L$, outputs the neighborhood $r$ that satisfies that maximum deviation. In other words, find the disc centered at $bar c$ with radius $g(y)$ such that every value of $f$ within that region is within $y$ of $L$. Symbolically:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x - bar c | < g(y) implies |f bar x - L| < y$$
Putting in the values for your problem:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x | < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
or to put more tersely, find $g : mathbb R to mathbb R$ such that for $bar x ne (0, 0)$
$$|bar x| < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
A polar substitution helps:
$$r < g(y) implies left|frac{(rcos t)^3 - (rsin t)^3}{(rcos t)^2 + (rsin t)^2}right| < y$$
Can you find the $g$ now? One more hint if you need it:
$-1 le cos t, sin t le 1$
add a comment |
up vote
0
down vote
$$(forall y > 0 ~ exists r > 0 ~ forall bar x )~ 0 < | bar x - bar c | < r implies |f bar x - L| < y$$
$(forall y > 0 ~ exists r > 0 ~ forall bar x )$ is equivalent to asking for the existence of a function $g$ that, given a maximum deviation from $L$, outputs the neighborhood $r$ that satisfies that maximum deviation. In other words, find the disc centered at $bar c$ with radius $g(y)$ such that every value of $f$ within that region is within $y$ of $L$. Symbolically:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x - bar c | < g(y) implies |f bar x - L| < y$$
Putting in the values for your problem:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x | < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
or to put more tersely, find $g : mathbb R to mathbb R$ such that for $bar x ne (0, 0)$
$$|bar x| < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
A polar substitution helps:
$$r < g(y) implies left|frac{(rcos t)^3 - (rsin t)^3}{(rcos t)^2 + (rsin t)^2}right| < y$$
Can you find the $g$ now? One more hint if you need it:
$-1 le cos t, sin t le 1$
add a comment |
up vote
0
down vote
up vote
0
down vote
$$(forall y > 0 ~ exists r > 0 ~ forall bar x )~ 0 < | bar x - bar c | < r implies |f bar x - L| < y$$
$(forall y > 0 ~ exists r > 0 ~ forall bar x )$ is equivalent to asking for the existence of a function $g$ that, given a maximum deviation from $L$, outputs the neighborhood $r$ that satisfies that maximum deviation. In other words, find the disc centered at $bar c$ with radius $g(y)$ such that every value of $f$ within that region is within $y$ of $L$. Symbolically:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x - bar c | < g(y) implies |f bar x - L| < y$$
Putting in the values for your problem:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x | < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
or to put more tersely, find $g : mathbb R to mathbb R$ such that for $bar x ne (0, 0)$
$$|bar x| < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
A polar substitution helps:
$$r < g(y) implies left|frac{(rcos t)^3 - (rsin t)^3}{(rcos t)^2 + (rsin t)^2}right| < y$$
Can you find the $g$ now? One more hint if you need it:
$-1 le cos t, sin t le 1$
$$(forall y > 0 ~ exists r > 0 ~ forall bar x )~ 0 < | bar x - bar c | < r implies |f bar x - L| < y$$
$(forall y > 0 ~ exists r > 0 ~ forall bar x )$ is equivalent to asking for the existence of a function $g$ that, given a maximum deviation from $L$, outputs the neighborhood $r$ that satisfies that maximum deviation. In other words, find the disc centered at $bar c$ with radius $g(y)$ such that every value of $f$ within that region is within $y$ of $L$. Symbolically:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x - bar c | < g(y) implies |f bar x - L| < y$$
Putting in the values for your problem:
$$(exists g : mathbb R to mathbb R ~forall y > 0 ~ forall bar x) ~ 0 < | bar x | < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
or to put more tersely, find $g : mathbb R to mathbb R$ such that for $bar x ne (0, 0)$
$$|bar x| < g(y) implies left|frac{x_1{}^3 - x_2{}^3}{x_1{}^2 + x_2{}^2}right| < y$$
A polar substitution helps:
$$r < g(y) implies left|frac{(rcos t)^3 - (rsin t)^3}{(rcos t)^2 + (rsin t)^2}right| < y$$
Can you find the $g$ now? One more hint if you need it:
$-1 le cos t, sin t le 1$
answered Nov 21 at 18:19
DanielV
17.8k42754
17.8k42754
add a comment |
add a comment |
1
This one is asking for it "by definition" whereas the other answers use shortcuts.
– DanielV
Nov 21 at 18:20