Show that the limit $lim_{(x,y)to(0,0)}frac{y+sin3x-3x}{y+x^5}$ exist/does not exist.
How do I show that this limit exist/does not exist? My assumption is that it does not; however I don't see how I can apply the two-path test to verify this.
$$lim_{(x,y)to(0,0)}dfrac{y+sin3x-3x}{y+x^5} = L,$$
$$lim_{yto0}dfrac{y+sin(0)-(0)}{y+(0)^5} = 1,$$
But I don't see any second path that simplifies this well;
$$lim_{xto0}dfrac{(0)+sin x-x}{(0)+x^5} = :?$$
calculus real-analysis limits
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
add a comment |
How do I show that this limit exist/does not exist? My assumption is that it does not; however I don't see how I can apply the two-path test to verify this.
$$lim_{(x,y)to(0,0)}dfrac{y+sin3x-3x}{y+x^5} = L,$$
$$lim_{yto0}dfrac{y+sin(0)-(0)}{y+(0)^5} = 1,$$
But I don't see any second path that simplifies this well;
$$lim_{xto0}dfrac{(0)+sin x-x}{(0)+x^5} = :?$$
calculus real-analysis limits
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
1
The denominator can be zero for points arbitrarily near to $(0,0)$.
– Martín-Blas Pérez Pinilla
Jan 29 '16 at 10:27
add a comment |
How do I show that this limit exist/does not exist? My assumption is that it does not; however I don't see how I can apply the two-path test to verify this.
$$lim_{(x,y)to(0,0)}dfrac{y+sin3x-3x}{y+x^5} = L,$$
$$lim_{yto0}dfrac{y+sin(0)-(0)}{y+(0)^5} = 1,$$
But I don't see any second path that simplifies this well;
$$lim_{xto0}dfrac{(0)+sin x-x}{(0)+x^5} = :?$$
calculus real-analysis limits
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
How do I show that this limit exist/does not exist? My assumption is that it does not; however I don't see how I can apply the two-path test to verify this.
$$lim_{(x,y)to(0,0)}dfrac{y+sin3x-3x}{y+x^5} = L,$$
$$lim_{yto0}dfrac{y+sin(0)-(0)}{y+(0)^5} = 1,$$
But I don't see any second path that simplifies this well;
$$lim_{xto0}dfrac{(0)+sin x-x}{(0)+x^5} = :?$$
calculus real-analysis limits
calculus real-analysis limits
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
edited Nov 28 '18 at 12:30
Lorenzo B.
1,8322520
1,8322520
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
asked Jan 29 '16 at 10:17
Frank Vel
2,34942246
2,34942246
1
The denominator can be zero for points arbitrarily near to $(0,0)$.
– Martín-Blas Pérez Pinilla
Jan 29 '16 at 10:27
add a comment |
1
The denominator can be zero for points arbitrarily near to $(0,0)$.
– Martín-Blas Pérez Pinilla
Jan 29 '16 at 10:27
1
1
The denominator can be zero for points arbitrarily near to $(0,0)$.
– Martín-Blas Pérez Pinilla
Jan 29 '16 at 10:27
The denominator can be zero for points arbitrarily near to $(0,0)$.
– Martín-Blas Pérez Pinilla
Jan 29 '16 at 10:27
add a comment |
1 Answer
1
active
oldest
votes
Hint
Let $f(x,y)=frac{y+sin(3x)-3x}{y+x^5}$.
What do you think about $$lim_{tto 0}f(0,t)quad text{and}quad lim_{tto 0}f(t,0) ?$$
Edited
$$sin(x)=x-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)$$
therefore $$frac{sin(x)-x}{x^5}=frac{-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)}{x^5}underset{xto 0}{longrightarrow }-infty .$$
answered Jan 29 '16 at 10:19
Surb
37.4k94375
Am I allowed to use L'Hôpital on the second one?
– Frank Vel
Jan 29 '16 at 10:22
1
It's indeed a possibility.
– Surb
Jan 29 '16 at 10:23
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1631799%2fshow-that-the-limit-lim-x-y-to0-0-fracy-sin3x-3xyx5-exist-does-n%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint
Let $f(x,y)=frac{y+sin(3x)-3x}{y+x^5}$.
What do you think about $$lim_{tto 0}f(0,t)quad text{and}quad lim_{tto 0}f(t,0) ?$$
Edited
$$sin(x)=x-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)$$
therefore $$frac{sin(x)-x}{x^5}=frac{-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)}{x^5}underset{xto 0}{longrightarrow }-infty .$$
answered Jan 29 '16 at 10:19
Surb
37.4k94375
Am I allowed to use L'Hôpital on the second one?
– Frank Vel
Jan 29 '16 at 10:22
1
It's indeed a possibility.
– Surb
Jan 29 '16 at 10:23
add a comment |
Hint
Let $f(x,y)=frac{y+sin(3x)-3x}{y+x^5}$.
What do you think about $$lim_{tto 0}f(0,t)quad text{and}quad lim_{tto 0}f(t,0) ?$$
Edited
$$sin(x)=x-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)$$
therefore $$frac{sin(x)-x}{x^5}=frac{-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)}{x^5}underset{xto 0}{longrightarrow }-infty .$$
answered Jan 29 '16 at 10:19
Surb
37.4k94375
Am I allowed to use L'Hôpital on the second one?
– Frank Vel
Jan 29 '16 at 10:22
1
It's indeed a possibility.
– Surb
Jan 29 '16 at 10:23
add a comment |
Hint
Let $f(x,y)=frac{y+sin(3x)-3x}{y+x^5}$.
What do you think about $$lim_{tto 0}f(0,t)quad text{and}quad lim_{tto 0}f(t,0) ?$$
Edited
$$sin(x)=x-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)$$
therefore $$frac{sin(x)-x}{x^5}=frac{-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)}{x^5}underset{xto 0}{longrightarrow }-infty .$$
answered Jan 29 '16 at 10:19
Surb
37.4k94375
Hint
Let $f(x,y)=frac{y+sin(3x)-3x}{y+x^5}$.
What do you think about $$lim_{tto 0}f(0,t)quad text{and}quad lim_{tto 0}f(t,0) ?$$
Edited
$$sin(x)=x-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)$$
therefore $$frac{sin(x)-x}{x^5}=frac{-frac{x^3}{3!}+frac{x^5}{5!}+o(x^5)}{x^5}underset{xto 0}{longrightarrow }-infty .$$
answered Jan 29 '16 at 10:19
Surb
37.4k94375
edited Jan 29 '16 at 10:23
answered Jan 29 '16 at 10:19
Surb
37.4k94375
answered Jan 29 '16 at 10:19
Surb
37.4k94375
answered Jan 29 '16 at 10:19
Surb
37.4k94375
37.4k94375
Am I allowed to use L'Hôpital on the second one?
– Frank Vel
Jan 29 '16 at 10:22
1
It's indeed a possibility.
– Surb
Jan 29 '16 at 10:23
add a comment |
Am I allowed to use L'Hôpital on the second one?
– Frank Vel
Jan 29 '16 at 10:22
1
It's indeed a possibility.
– Surb
Jan 29 '16 at 10:23
Am I allowed to use L'Hôpital on the second one?
– Frank Vel
Jan 29 '16 at 10:22
Am I allowed to use L'Hôpital on the second one?
– Frank Vel
Jan 29 '16 at 10:22
1
1
It's indeed a possibility.
– Surb
Jan 29 '16 at 10:23
It's indeed a possibility.
– Surb
Jan 29 '16 at 10:23
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1631799%2fshow-that-the-limit-lim-x-y-to0-0-fracy-sin3x-3xyx5-exist-does-n%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
The denominator can be zero for points arbitrarily near to $(0,0)$.
– Martín-Blas Pérez Pinilla
Jan 29 '16 at 10:27