Spivak, Differentiability (An Exercise)












0














I am working on a problem from Spivak, question 52b. I will type part a for reference as well since it may be relevant to the problem at hand. It is stated as follows:




There is another form of L'Hôpital's Rule which requires more than algebraic
manipulations: If $lim_{xrightarrow infty}f(x)=lim_{xrightarrow infty}g(x)=infty$ and $lim_{xrightarrow infty}frac{f'(x)}{g'(x)}=l$ then $lim_{xrightarrow infty}frac{f(x)}{g(x)}=l$. Prove this as follows.



(a) For every $varepsilon>0$ there is a number $a$ such that $$left|frac{f'(x)}{g'(x)}-lright| <varepsilonqquadtext{for }x>a.$$
Apply the Cauchy Mean Value Theorem to $f$ and $g$ on $[a,x]$ to show that $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilonqquadtext{for }x>a.$$
(Why can we assume that $g(x)-g(a) not=0?$)



(b) Now write
$$frac{f(x)}{g(x)}=frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}$$
(why can we assume that $f(x)-f(a)not=0$ for large x?) and conclude that
$$left|frac{f(x)}{g(x)}-lright|<2varepsilonquadtext{for sufficiently large }x.$$




For part b, which is where my troubles lie, I have dealt with the limits $lim_{xrightarrow infty}frac{f(x)}{f(x)-f(a)}$ and $lim_{xrightarrow infty}frac{g(x)-g(a)}{g(x)}$ and I was able to conclude both are 1, and the limit $lim_{xrightarrow infty}frac{f(x)-f(a)}{g(x)-g(a)}$ was dealt with in part a via Cauchy MVT. And we know that for large enough $x$ (say, $x>N$), $f$ is unbounded so we can infer that $f(x)-f(a) not=0$ for large enough $x$. Now this is all fine, but I cannot for the life of me figure out where a $2epsilon$ comes from. I have been playing with this for a couple days with no luck.



Thanks ahead of time for any help.










share|cite|improve this question
























  • Well note that by assumption $g'(x) $ is non-zero after a certain value of $x$ say $a$ and by Darboux theorem it is of constant sign. Thus $g$ is strictly monotone in $[a, infty) $ and that means $g(x) neq g(a)$ if $x>a$. The other part is mostly the typical $epsilon, delta$ gymnastics which involves triangle inequality and some simple bounds for various terms.
    – Paramanand Singh
    Dec 1 at 15:50
















0














I am working on a problem from Spivak, question 52b. I will type part a for reference as well since it may be relevant to the problem at hand. It is stated as follows:




There is another form of L'Hôpital's Rule which requires more than algebraic
manipulations: If $lim_{xrightarrow infty}f(x)=lim_{xrightarrow infty}g(x)=infty$ and $lim_{xrightarrow infty}frac{f'(x)}{g'(x)}=l$ then $lim_{xrightarrow infty}frac{f(x)}{g(x)}=l$. Prove this as follows.



(a) For every $varepsilon>0$ there is a number $a$ such that $$left|frac{f'(x)}{g'(x)}-lright| <varepsilonqquadtext{for }x>a.$$
Apply the Cauchy Mean Value Theorem to $f$ and $g$ on $[a,x]$ to show that $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilonqquadtext{for }x>a.$$
(Why can we assume that $g(x)-g(a) not=0?$)



(b) Now write
$$frac{f(x)}{g(x)}=frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}$$
(why can we assume that $f(x)-f(a)not=0$ for large x?) and conclude that
$$left|frac{f(x)}{g(x)}-lright|<2varepsilonquadtext{for sufficiently large }x.$$




For part b, which is where my troubles lie, I have dealt with the limits $lim_{xrightarrow infty}frac{f(x)}{f(x)-f(a)}$ and $lim_{xrightarrow infty}frac{g(x)-g(a)}{g(x)}$ and I was able to conclude both are 1, and the limit $lim_{xrightarrow infty}frac{f(x)-f(a)}{g(x)-g(a)}$ was dealt with in part a via Cauchy MVT. And we know that for large enough $x$ (say, $x>N$), $f$ is unbounded so we can infer that $f(x)-f(a) not=0$ for large enough $x$. Now this is all fine, but I cannot for the life of me figure out where a $2epsilon$ comes from. I have been playing with this for a couple days with no luck.



Thanks ahead of time for any help.










share|cite|improve this question
























  • Well note that by assumption $g'(x) $ is non-zero after a certain value of $x$ say $a$ and by Darboux theorem it is of constant sign. Thus $g$ is strictly monotone in $[a, infty) $ and that means $g(x) neq g(a)$ if $x>a$. The other part is mostly the typical $epsilon, delta$ gymnastics which involves triangle inequality and some simple bounds for various terms.
    – Paramanand Singh
    Dec 1 at 15:50














0












0








0







I am working on a problem from Spivak, question 52b. I will type part a for reference as well since it may be relevant to the problem at hand. It is stated as follows:




There is another form of L'Hôpital's Rule which requires more than algebraic
manipulations: If $lim_{xrightarrow infty}f(x)=lim_{xrightarrow infty}g(x)=infty$ and $lim_{xrightarrow infty}frac{f'(x)}{g'(x)}=l$ then $lim_{xrightarrow infty}frac{f(x)}{g(x)}=l$. Prove this as follows.



(a) For every $varepsilon>0$ there is a number $a$ such that $$left|frac{f'(x)}{g'(x)}-lright| <varepsilonqquadtext{for }x>a.$$
Apply the Cauchy Mean Value Theorem to $f$ and $g$ on $[a,x]$ to show that $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilonqquadtext{for }x>a.$$
(Why can we assume that $g(x)-g(a) not=0?$)



(b) Now write
$$frac{f(x)}{g(x)}=frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}$$
(why can we assume that $f(x)-f(a)not=0$ for large x?) and conclude that
$$left|frac{f(x)}{g(x)}-lright|<2varepsilonquadtext{for sufficiently large }x.$$




For part b, which is where my troubles lie, I have dealt with the limits $lim_{xrightarrow infty}frac{f(x)}{f(x)-f(a)}$ and $lim_{xrightarrow infty}frac{g(x)-g(a)}{g(x)}$ and I was able to conclude both are 1, and the limit $lim_{xrightarrow infty}frac{f(x)-f(a)}{g(x)-g(a)}$ was dealt with in part a via Cauchy MVT. And we know that for large enough $x$ (say, $x>N$), $f$ is unbounded so we can infer that $f(x)-f(a) not=0$ for large enough $x$. Now this is all fine, but I cannot for the life of me figure out where a $2epsilon$ comes from. I have been playing with this for a couple days with no luck.



Thanks ahead of time for any help.










share|cite|improve this question















I am working on a problem from Spivak, question 52b. I will type part a for reference as well since it may be relevant to the problem at hand. It is stated as follows:




There is another form of L'Hôpital's Rule which requires more than algebraic
manipulations: If $lim_{xrightarrow infty}f(x)=lim_{xrightarrow infty}g(x)=infty$ and $lim_{xrightarrow infty}frac{f'(x)}{g'(x)}=l$ then $lim_{xrightarrow infty}frac{f(x)}{g(x)}=l$. Prove this as follows.



(a) For every $varepsilon>0$ there is a number $a$ such that $$left|frac{f'(x)}{g'(x)}-lright| <varepsilonqquadtext{for }x>a.$$
Apply the Cauchy Mean Value Theorem to $f$ and $g$ on $[a,x]$ to show that $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilonqquadtext{for }x>a.$$
(Why can we assume that $g(x)-g(a) not=0?$)



(b) Now write
$$frac{f(x)}{g(x)}=frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}$$
(why can we assume that $f(x)-f(a)not=0$ for large x?) and conclude that
$$left|frac{f(x)}{g(x)}-lright|<2varepsilonquadtext{for sufficiently large }x.$$




For part b, which is where my troubles lie, I have dealt with the limits $lim_{xrightarrow infty}frac{f(x)}{f(x)-f(a)}$ and $lim_{xrightarrow infty}frac{g(x)-g(a)}{g(x)}$ and I was able to conclude both are 1, and the limit $lim_{xrightarrow infty}frac{f(x)-f(a)}{g(x)-g(a)}$ was dealt with in part a via Cauchy MVT. And we know that for large enough $x$ (say, $x>N$), $f$ is unbounded so we can infer that $f(x)-f(a) not=0$ for large enough $x$. Now this is all fine, but I cannot for the life of me figure out where a $2epsilon$ comes from. I have been playing with this for a couple days with no luck.



Thanks ahead of time for any help.







calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 1 at 10:33









Luiz Cordeiro

12.5k1143




12.5k1143










asked Dec 1 at 9:59









kyle campbell

244




244












  • Well note that by assumption $g'(x) $ is non-zero after a certain value of $x$ say $a$ and by Darboux theorem it is of constant sign. Thus $g$ is strictly monotone in $[a, infty) $ and that means $g(x) neq g(a)$ if $x>a$. The other part is mostly the typical $epsilon, delta$ gymnastics which involves triangle inequality and some simple bounds for various terms.
    – Paramanand Singh
    Dec 1 at 15:50


















  • Well note that by assumption $g'(x) $ is non-zero after a certain value of $x$ say $a$ and by Darboux theorem it is of constant sign. Thus $g$ is strictly monotone in $[a, infty) $ and that means $g(x) neq g(a)$ if $x>a$. The other part is mostly the typical $epsilon, delta$ gymnastics which involves triangle inequality and some simple bounds for various terms.
    – Paramanand Singh
    Dec 1 at 15:50
















Well note that by assumption $g'(x) $ is non-zero after a certain value of $x$ say $a$ and by Darboux theorem it is of constant sign. Thus $g$ is strictly monotone in $[a, infty) $ and that means $g(x) neq g(a)$ if $x>a$. The other part is mostly the typical $epsilon, delta$ gymnastics which involves triangle inequality and some simple bounds for various terms.
– Paramanand Singh
Dec 1 at 15:50




Well note that by assumption $g'(x) $ is non-zero after a certain value of $x$ say $a$ and by Darboux theorem it is of constant sign. Thus $g$ is strictly monotone in $[a, infty) $ and that means $g(x) neq g(a)$ if $x>a$. The other part is mostly the typical $epsilon, delta$ gymnastics which involves triangle inequality and some simple bounds for various terms.
– Paramanand Singh
Dec 1 at 15:50










1 Answer
1






active

oldest

votes


















1














Basically, write down whatevver the hint says. You will have an expression of the form $|A(x)-l|$ for some $A(x)$. Use the first part and write $|A(x)-l|leq |A(x)-frac{f(x)-f(a)}{g(x)-g(a)}|+|frac{f(x)-f(a)}{g(x)-g(a)}-l|$ and try to work from there. Below are the details.




begin{align*}
left|frac{f(x)}{g(x)}-lright|&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}+frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&leqleft|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}right|cdotleft|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|
end{align*}



The term $left|frac{f(x)-f(a)}{g(x)-g(a)}right|$ gets close to $|l|$ (so it may be bounded by $|l|+1$), the term mutiplying it gets close to $0$, and the last term gets close to $0$ as well, so I'll leave the details to you







share|cite|improve this answer





















  • Thank you! Forgive me, since I must be missing something obvious, but I am still not seeing how a 2$varepsilon$ pops out. Any hints?
    – kyle campbell
    Dec 1 at 20:56






  • 1




    For large $x$, we have $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<1,$$ so $$left|frac{f(x)-f(a)}{g(x)-g(a)}right|leq |l|+1.$$ For $x$ sufficiently large, we also have $$left|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|<varepsilon/2(|l|+1)$$ and $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilon/2$$ so substituting all these inequalities in the large equation in the answer above gives the term $2varepsilon$. The key is that item (a) holds for any $varepsilon$, so in particular for anything of the form $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:14








  • 1




    (cont.) How large $x$ needs to be will depend, however, on your given $varepsilon$, $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:17











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%2f3021182%2fspivak-differentiability-an-exercise%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









1














Basically, write down whatevver the hint says. You will have an expression of the form $|A(x)-l|$ for some $A(x)$. Use the first part and write $|A(x)-l|leq |A(x)-frac{f(x)-f(a)}{g(x)-g(a)}|+|frac{f(x)-f(a)}{g(x)-g(a)}-l|$ and try to work from there. Below are the details.




begin{align*}
left|frac{f(x)}{g(x)}-lright|&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}+frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&leqleft|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}right|cdotleft|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|
end{align*}



The term $left|frac{f(x)-f(a)}{g(x)-g(a)}right|$ gets close to $|l|$ (so it may be bounded by $|l|+1$), the term mutiplying it gets close to $0$, and the last term gets close to $0$ as well, so I'll leave the details to you







share|cite|improve this answer





















  • Thank you! Forgive me, since I must be missing something obvious, but I am still not seeing how a 2$varepsilon$ pops out. Any hints?
    – kyle campbell
    Dec 1 at 20:56






  • 1




    For large $x$, we have $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<1,$$ so $$left|frac{f(x)-f(a)}{g(x)-g(a)}right|leq |l|+1.$$ For $x$ sufficiently large, we also have $$left|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|<varepsilon/2(|l|+1)$$ and $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilon/2$$ so substituting all these inequalities in the large equation in the answer above gives the term $2varepsilon$. The key is that item (a) holds for any $varepsilon$, so in particular for anything of the form $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:14








  • 1




    (cont.) How large $x$ needs to be will depend, however, on your given $varepsilon$, $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:17
















1














Basically, write down whatevver the hint says. You will have an expression of the form $|A(x)-l|$ for some $A(x)$. Use the first part and write $|A(x)-l|leq |A(x)-frac{f(x)-f(a)}{g(x)-g(a)}|+|frac{f(x)-f(a)}{g(x)-g(a)}-l|$ and try to work from there. Below are the details.




begin{align*}
left|frac{f(x)}{g(x)}-lright|&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}+frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&leqleft|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}right|cdotleft|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|
end{align*}



The term $left|frac{f(x)-f(a)}{g(x)-g(a)}right|$ gets close to $|l|$ (so it may be bounded by $|l|+1$), the term mutiplying it gets close to $0$, and the last term gets close to $0$ as well, so I'll leave the details to you







share|cite|improve this answer





















  • Thank you! Forgive me, since I must be missing something obvious, but I am still not seeing how a 2$varepsilon$ pops out. Any hints?
    – kyle campbell
    Dec 1 at 20:56






  • 1




    For large $x$, we have $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<1,$$ so $$left|frac{f(x)-f(a)}{g(x)-g(a)}right|leq |l|+1.$$ For $x$ sufficiently large, we also have $$left|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|<varepsilon/2(|l|+1)$$ and $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilon/2$$ so substituting all these inequalities in the large equation in the answer above gives the term $2varepsilon$. The key is that item (a) holds for any $varepsilon$, so in particular for anything of the form $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:14








  • 1




    (cont.) How large $x$ needs to be will depend, however, on your given $varepsilon$, $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:17














1












1








1






Basically, write down whatevver the hint says. You will have an expression of the form $|A(x)-l|$ for some $A(x)$. Use the first part and write $|A(x)-l|leq |A(x)-frac{f(x)-f(a)}{g(x)-g(a)}|+|frac{f(x)-f(a)}{g(x)-g(a)}-l|$ and try to work from there. Below are the details.




begin{align*}
left|frac{f(x)}{g(x)}-lright|&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}+frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&leqleft|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}right|cdotleft|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|
end{align*}



The term $left|frac{f(x)-f(a)}{g(x)-g(a)}right|$ gets close to $|l|$ (so it may be bounded by $|l|+1$), the term mutiplying it gets close to $0$, and the last term gets close to $0$ as well, so I'll leave the details to you







share|cite|improve this answer












Basically, write down whatevver the hint says. You will have an expression of the form $|A(x)-l|$ for some $A(x)$. Use the first part and write $|A(x)-l|leq |A(x)-frac{f(x)-f(a)}{g(x)-g(a)}|+|frac{f(x)-f(a)}{g(x)-g(a)}-l|$ and try to work from there. Below are the details.




begin{align*}
left|frac{f(x)}{g(x)}-lright|&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}+frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&leqleft|frac{f(x)-f(a)}{g(x)-g(a)}cdotfrac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-frac{f(x)-f(a)}{g(x)-g(a)}right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|\
&=left|frac{f(x)-f(a)}{g(x)-g(a)}right|cdotleft|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|+left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|
end{align*}



The term $left|frac{f(x)-f(a)}{g(x)-g(a)}right|$ gets close to $|l|$ (so it may be bounded by $|l|+1$), the term mutiplying it gets close to $0$, and the last term gets close to $0$ as well, so I'll leave the details to you








share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 1 at 10:47









Luiz Cordeiro

12.5k1143




12.5k1143












  • Thank you! Forgive me, since I must be missing something obvious, but I am still not seeing how a 2$varepsilon$ pops out. Any hints?
    – kyle campbell
    Dec 1 at 20:56






  • 1




    For large $x$, we have $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<1,$$ so $$left|frac{f(x)-f(a)}{g(x)-g(a)}right|leq |l|+1.$$ For $x$ sufficiently large, we also have $$left|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|<varepsilon/2(|l|+1)$$ and $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilon/2$$ so substituting all these inequalities in the large equation in the answer above gives the term $2varepsilon$. The key is that item (a) holds for any $varepsilon$, so in particular for anything of the form $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:14








  • 1




    (cont.) How large $x$ needs to be will depend, however, on your given $varepsilon$, $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:17


















  • Thank you! Forgive me, since I must be missing something obvious, but I am still not seeing how a 2$varepsilon$ pops out. Any hints?
    – kyle campbell
    Dec 1 at 20:56






  • 1




    For large $x$, we have $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<1,$$ so $$left|frac{f(x)-f(a)}{g(x)-g(a)}right|leq |l|+1.$$ For $x$ sufficiently large, we also have $$left|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|<varepsilon/2(|l|+1)$$ and $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilon/2$$ so substituting all these inequalities in the large equation in the answer above gives the term $2varepsilon$. The key is that item (a) holds for any $varepsilon$, so in particular for anything of the form $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:14








  • 1




    (cont.) How large $x$ needs to be will depend, however, on your given $varepsilon$, $varepsilon/2$, $varepsilon^2$, etc...
    – Luiz Cordeiro
    Dec 2 at 0:17
















Thank you! Forgive me, since I must be missing something obvious, but I am still not seeing how a 2$varepsilon$ pops out. Any hints?
– kyle campbell
Dec 1 at 20:56




Thank you! Forgive me, since I must be missing something obvious, but I am still not seeing how a 2$varepsilon$ pops out. Any hints?
– kyle campbell
Dec 1 at 20:56




1




1




For large $x$, we have $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<1,$$ so $$left|frac{f(x)-f(a)}{g(x)-g(a)}right|leq |l|+1.$$ For $x$ sufficiently large, we also have $$left|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|<varepsilon/2(|l|+1)$$ and $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilon/2$$ so substituting all these inequalities in the large equation in the answer above gives the term $2varepsilon$. The key is that item (a) holds for any $varepsilon$, so in particular for anything of the form $varepsilon/2$, $varepsilon^2$, etc...
– Luiz Cordeiro
Dec 2 at 0:14






For large $x$, we have $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<1,$$ so $$left|frac{f(x)-f(a)}{g(x)-g(a)}right|leq |l|+1.$$ For $x$ sufficiently large, we also have $$left|frac{f(x)}{f(x)-f(a)}cdotfrac{g(x)-g(a)}{g(x)}-1right|<varepsilon/2(|l|+1)$$ and $$left|frac{f(x)-f(a)}{g(x)-g(a)}-lright|<varepsilon/2$$ so substituting all these inequalities in the large equation in the answer above gives the term $2varepsilon$. The key is that item (a) holds for any $varepsilon$, so in particular for anything of the form $varepsilon/2$, $varepsilon^2$, etc...
– Luiz Cordeiro
Dec 2 at 0:14






1




1




(cont.) How large $x$ needs to be will depend, however, on your given $varepsilon$, $varepsilon/2$, $varepsilon^2$, etc...
– Luiz Cordeiro
Dec 2 at 0:17




(cont.) How large $x$ needs to be will depend, however, on your given $varepsilon$, $varepsilon/2$, $varepsilon^2$, etc...
– Luiz Cordeiro
Dec 2 at 0:17


















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.





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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3021182%2fspivak-differentiability-an-exercise%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

Berounka

Sphinx de Gizeh

Different font size/position of beamer's navigation symbols template's content depending on regular/plain...