prove that the limit of the given piecewise function does not exist, using the formal definition.












2












$begingroup$


i'm having problem with this exercise. Can somebody give me a hint ?



The function is
f(x) =
begin{cases}
x^{2}, & text{if x ≤ 2 } \
8 - 2x, & text{if 2 < x}
end{cases}



is given that the $limlimits_{x to 2} f(x)$ does not exist. but how i prove with the formal definition ?



i use the definition ∀ε > 0 ∃δ > 0 : | x ∋ (a,b), 0<|x-x'|<δ and |f(x) - L|<ε



and tried, first fixing ε = 0.1 , x'= 2 then applying |1.9 - 2|< δ => δ > 0.1 and |4.2 - 4| < ε => ε > 0.2 so by this result the limit exist ?!
Thanks.



edit: I misunderstood the concept of the existence of a limit and if the function can be derived. then yes the limit exist. But still in my professor problem are saying that the limit does not exist. Translating it is says literaly "prove by the formal definition why the lim does not exist". Probably he was playing with the class to see what we would answer.






calculus limits







share|cite|improve this question











$endgroup$












  • $begingroup$
    Unfortunately, during translation something seems to have been lost. The limit exists at $x = 2$, and in fact this function is continuous everywhere, including at $2$.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:03










  • $begingroup$
    you are right, i bet the professor was probably playing with us making this kind of affirmation in the problem. Just to see what we would answer.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:05










  • $begingroup$
    I see. Also, to close the question, either write an answer yourself and accept it after some time, or accept the answer below when you can (if you like it, but there is everything to like about it): this will automatically close the question.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:06












  • $begingroup$
    did it, thanks :)
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:08










  • $begingroup$
    You are welcome! +1 for the question, and providing attention to answers/comments.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:08


















2












$begingroup$


i'm having problem with this exercise. Can somebody give me a hint ?



The function is
f(x) =
begin{cases}
x^{2}, & text{if x ≤ 2 } \
8 - 2x, & text{if 2 < x}
end{cases}



is given that the $limlimits_{x to 2} f(x)$ does not exist. but how i prove with the formal definition ?



i use the definition ∀ε > 0 ∃δ > 0 : | x ∋ (a,b), 0<|x-x'|<δ and |f(x) - L|<ε



and tried, first fixing ε = 0.1 , x'= 2 then applying |1.9 - 2|< δ => δ > 0.1 and |4.2 - 4| < ε => ε > 0.2 so by this result the limit exist ?!
Thanks.



edit: I misunderstood the concept of the existence of a limit and if the function can be derived. then yes the limit exist. But still in my professor problem are saying that the limit does not exist. Translating it is says literaly "prove by the formal definition why the lim does not exist". Probably he was playing with the class to see what we would answer.






calculus limits







share|cite|improve this question











$endgroup$












  • $begingroup$
    Unfortunately, during translation something seems to have been lost. The limit exists at $x = 2$, and in fact this function is continuous everywhere, including at $2$.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:03










  • $begingroup$
    you are right, i bet the professor was probably playing with us making this kind of affirmation in the problem. Just to see what we would answer.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:05










  • $begingroup$
    I see. Also, to close the question, either write an answer yourself and accept it after some time, or accept the answer below when you can (if you like it, but there is everything to like about it): this will automatically close the question.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:06












  • $begingroup$
    did it, thanks :)
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:08










  • $begingroup$
    You are welcome! +1 for the question, and providing attention to answers/comments.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:08
















2












2








2





$begingroup$


i'm having problem with this exercise. Can somebody give me a hint ?



The function is
f(x) =
begin{cases}
x^{2}, & text{if x ≤ 2 } \
8 - 2x, & text{if 2 < x}
end{cases}



is given that the $limlimits_{x to 2} f(x)$ does not exist. but how i prove with the formal definition ?



i use the definition ∀ε > 0 ∃δ > 0 : | x ∋ (a,b), 0<|x-x'|<δ and |f(x) - L|<ε



and tried, first fixing ε = 0.1 , x'= 2 then applying |1.9 - 2|< δ => δ > 0.1 and |4.2 - 4| < ε => ε > 0.2 so by this result the limit exist ?!
Thanks.



edit: I misunderstood the concept of the existence of a limit and if the function can be derived. then yes the limit exist. But still in my professor problem are saying that the limit does not exist. Translating it is says literaly "prove by the formal definition why the lim does not exist". Probably he was playing with the class to see what we would answer.






calculus limits







share|cite|improve this question











$endgroup$




i'm having problem with this exercise. Can somebody give me a hint ?



The function is
f(x) =
begin{cases}
x^{2}, & text{if x ≤ 2 } \
8 - 2x, & text{if 2 < x}
end{cases}



is given that the $limlimits_{x to 2} f(x)$ does not exist. but how i prove with the formal definition ?



i use the definition ∀ε > 0 ∃δ > 0 : | x ∋ (a,b), 0<|x-x'|<δ and |f(x) - L|<ε



and tried, first fixing ε = 0.1 , x'= 2 then applying |1.9 - 2|< δ => δ > 0.1 and |4.2 - 4| < ε => ε > 0.2 so by this result the limit exist ?!
Thanks.



edit: I misunderstood the concept of the existence of a limit and if the function can be derived. then yes the limit exist. But still in my professor problem are saying that the limit does not exist. Translating it is says literaly "prove by the formal definition why the lim does not exist". Probably he was playing with the class to see what we would answer.






calculus limits




calculus limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 16:11







Jeremias Junior

















asked Dec 5 '18 at 15:45









Jeremias JuniorJeremias Junior

154




154












  • $begingroup$
    Unfortunately, during translation something seems to have been lost. The limit exists at $x = 2$, and in fact this function is continuous everywhere, including at $2$.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:03










  • $begingroup$
    you are right, i bet the professor was probably playing with us making this kind of affirmation in the problem. Just to see what we would answer.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:05










  • $begingroup$
    I see. Also, to close the question, either write an answer yourself and accept it after some time, or accept the answer below when you can (if you like it, but there is everything to like about it): this will automatically close the question.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:06












  • $begingroup$
    did it, thanks :)
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:08










  • $begingroup$
    You are welcome! +1 for the question, and providing attention to answers/comments.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:08




















  • $begingroup$
    Unfortunately, during translation something seems to have been lost. The limit exists at $x = 2$, and in fact this function is continuous everywhere, including at $2$.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:03










  • $begingroup$
    you are right, i bet the professor was probably playing with us making this kind of affirmation in the problem. Just to see what we would answer.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:05










  • $begingroup$
    I see. Also, to close the question, either write an answer yourself and accept it after some time, or accept the answer below when you can (if you like it, but there is everything to like about it): this will automatically close the question.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:06












  • $begingroup$
    did it, thanks :)
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:08










  • $begingroup$
    You are welcome! +1 for the question, and providing attention to answers/comments.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Dec 5 '18 at 16:08


















$begingroup$
Unfortunately, during translation something seems to have been lost. The limit exists at $x = 2$, and in fact this function is continuous everywhere, including at $2$.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 5 '18 at 16:03




$begingroup$
Unfortunately, during translation something seems to have been lost. The limit exists at $x = 2$, and in fact this function is continuous everywhere, including at $2$.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 5 '18 at 16:03












$begingroup$
you are right, i bet the professor was probably playing with us making this kind of affirmation in the problem. Just to see what we would answer.
$endgroup$
– Jeremias Junior
Dec 5 '18 at 16:05




$begingroup$
you are right, i bet the professor was probably playing with us making this kind of affirmation in the problem. Just to see what we would answer.
$endgroup$
– Jeremias Junior
Dec 5 '18 at 16:05












$begingroup$
I see. Also, to close the question, either write an answer yourself and accept it after some time, or accept the answer below when you can (if you like it, but there is everything to like about it): this will automatically close the question.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 5 '18 at 16:06






$begingroup$
I see. Also, to close the question, either write an answer yourself and accept it after some time, or accept the answer below when you can (if you like it, but there is everything to like about it): this will automatically close the question.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 5 '18 at 16:06














$begingroup$
did it, thanks :)
$endgroup$
– Jeremias Junior
Dec 5 '18 at 16:08




$begingroup$
did it, thanks :)
$endgroup$
– Jeremias Junior
Dec 5 '18 at 16:08












$begingroup$
You are welcome! +1 for the question, and providing attention to answers/comments.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 5 '18 at 16:08






$begingroup$
You are welcome! +1 for the question, and providing attention to answers/comments.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 5 '18 at 16:08












1 Answer
1






active

oldest

votes


















0












$begingroup$

Note that indeed



$$limlimits_{x to 2^+} f(x)=limlimits_{x to 2^-} f(x)=4$$



therefore the limit exists and you can proceed with the definition to prove that.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Oh i made a huge mistake! on my interpretation.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:51












  • $begingroup$
    translating it is says literaly "prove by the formal definition why the lim does not exist".
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:57










  • $begingroup$
    @JeremiasJunior Yes but here the limit exists, there we can't prove that!
    $endgroup$
    – gimusi
    Dec 5 '18 at 15:58










  • $begingroup$
    for sure he playing with the class, by giving this kind of affirmation. Thanks again.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:01










  • $begingroup$
    Maybe it is a typo for the question or for the function. You can prove at first by definition that it exists and the taking $f(x)=8-3x$ for $x>2$ prove by definiiton that it doesn't exist.
    $endgroup$
    – gimusi
    Dec 5 '18 at 16:02











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027221%2fprove-that-the-limit-of-the-given-piecewise-function-does-not-exist-using-the-f%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









0












$begingroup$

Note that indeed



$$limlimits_{x to 2^+} f(x)=limlimits_{x to 2^-} f(x)=4$$



therefore the limit exists and you can proceed with the definition to prove that.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Oh i made a huge mistake! on my interpretation.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:51












  • $begingroup$
    translating it is says literaly "prove by the formal definition why the lim does not exist".
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:57










  • $begingroup$
    @JeremiasJunior Yes but here the limit exists, there we can't prove that!
    $endgroup$
    – gimusi
    Dec 5 '18 at 15:58










  • $begingroup$
    for sure he playing with the class, by giving this kind of affirmation. Thanks again.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:01










  • $begingroup$
    Maybe it is a typo for the question or for the function. You can prove at first by definition that it exists and the taking $f(x)=8-3x$ for $x>2$ prove by definiiton that it doesn't exist.
    $endgroup$
    – gimusi
    Dec 5 '18 at 16:02
















0












$begingroup$

Note that indeed



$$limlimits_{x to 2^+} f(x)=limlimits_{x to 2^-} f(x)=4$$



therefore the limit exists and you can proceed with the definition to prove that.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Oh i made a huge mistake! on my interpretation.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:51












  • $begingroup$
    translating it is says literaly "prove by the formal definition why the lim does not exist".
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:57










  • $begingroup$
    @JeremiasJunior Yes but here the limit exists, there we can't prove that!
    $endgroup$
    – gimusi
    Dec 5 '18 at 15:58










  • $begingroup$
    for sure he playing with the class, by giving this kind of affirmation. Thanks again.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:01










  • $begingroup$
    Maybe it is a typo for the question or for the function. You can prove at first by definition that it exists and the taking $f(x)=8-3x$ for $x>2$ prove by definiiton that it doesn't exist.
    $endgroup$
    – gimusi
    Dec 5 '18 at 16:02














0












0








0





$begingroup$

Note that indeed



$$limlimits_{x to 2^+} f(x)=limlimits_{x to 2^-} f(x)=4$$



therefore the limit exists and you can proceed with the definition to prove that.






share|cite|improve this answer









$endgroup$



Note that indeed



$$limlimits_{x to 2^+} f(x)=limlimits_{x to 2^-} f(x)=4$$



therefore the limit exists and you can proceed with the definition to prove that.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 5 '18 at 15:48









gimusigimusi

92.9k84494




92.9k84494












  • $begingroup$
    Oh i made a huge mistake! on my interpretation.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:51












  • $begingroup$
    translating it is says literaly "prove by the formal definition why the lim does not exist".
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:57










  • $begingroup$
    @JeremiasJunior Yes but here the limit exists, there we can't prove that!
    $endgroup$
    – gimusi
    Dec 5 '18 at 15:58










  • $begingroup$
    for sure he playing with the class, by giving this kind of affirmation. Thanks again.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:01










  • $begingroup$
    Maybe it is a typo for the question or for the function. You can prove at first by definition that it exists and the taking $f(x)=8-3x$ for $x>2$ prove by definiiton that it doesn't exist.
    $endgroup$
    – gimusi
    Dec 5 '18 at 16:02


















  • $begingroup$
    Oh i made a huge mistake! on my interpretation.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:51












  • $begingroup$
    translating it is says literaly "prove by the formal definition why the lim does not exist".
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 15:57










  • $begingroup$
    @JeremiasJunior Yes but here the limit exists, there we can't prove that!
    $endgroup$
    – gimusi
    Dec 5 '18 at 15:58










  • $begingroup$
    for sure he playing with the class, by giving this kind of affirmation. Thanks again.
    $endgroup$
    – Jeremias Junior
    Dec 5 '18 at 16:01










  • $begingroup$
    Maybe it is a typo for the question or for the function. You can prove at first by definition that it exists and the taking $f(x)=8-3x$ for $x>2$ prove by definiiton that it doesn't exist.
    $endgroup$
    – gimusi
    Dec 5 '18 at 16:02
















$begingroup$
Oh i made a huge mistake! on my interpretation.
$endgroup$
– Jeremias Junior
Dec 5 '18 at 15:51






$begingroup$
Oh i made a huge mistake! on my interpretation.
$endgroup$
– Jeremias Junior
Dec 5 '18 at 15:51














$begingroup$
translating it is says literaly "prove by the formal definition why the lim does not exist".
$endgroup$
– Jeremias Junior
Dec 5 '18 at 15:57




$begingroup$
translating it is says literaly "prove by the formal definition why the lim does not exist".
$endgroup$
– Jeremias Junior
Dec 5 '18 at 15:57












$begingroup$
@JeremiasJunior Yes but here the limit exists, there we can't prove that!
$endgroup$
– gimusi
Dec 5 '18 at 15:58




$begingroup$
@JeremiasJunior Yes but here the limit exists, there we can't prove that!
$endgroup$
– gimusi
Dec 5 '18 at 15:58












$begingroup$
for sure he playing with the class, by giving this kind of affirmation. Thanks again.
$endgroup$
– Jeremias Junior
Dec 5 '18 at 16:01




$begingroup$
for sure he playing with the class, by giving this kind of affirmation. Thanks again.
$endgroup$
– Jeremias Junior
Dec 5 '18 at 16:01












$begingroup$
Maybe it is a typo for the question or for the function. You can prove at first by definition that it exists and the taking $f(x)=8-3x$ for $x>2$ prove by definiiton that it doesn't exist.
$endgroup$
– gimusi
Dec 5 '18 at 16:02




$begingroup$
Maybe it is a typo for the question or for the function. You can prove at first by definition that it exists and the taking $f(x)=8-3x$ for $x>2$ prove by definiiton that it doesn't exist.
$endgroup$
– gimusi
Dec 5 '18 at 16:02


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027221%2fprove-that-the-limit-of-the-given-piecewise-function-does-not-exist-using-the-f%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

How to change which sound is reproduced for terminal bell?

Image
3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
Read more

Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

Image
up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
Read more

Can I use Tabulator js library in my java Spring + Thymeleaf project?

Image
0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
Read more