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Excerpt from chapter on Differentiation(P.3) tells that

To say that $\lim_{h\to0}(2+h^2)$ doesn’t quite mean that $(2+h^2)$ gets closer and closer to $2$ as $h$ gets closer to $0$.After all $(2+h^2)$ gets closer and closer to 1 as h gets closer and closer to 0, but $\lim_{h\to0}(2+h^2)=2$,not 1.

Why does the article says that "$\lim_{h\to0}(2+h^2)$ doesn’t quite mean that $(2+h^2)$ gets closer and closer to $2$ as $h$ gets closer to $0$"?

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    $\begingroup$ I think that means that 2 is not the unique number $2+h^2$ get closer to when $h$ goes to $0$. $\endgroup$ – nicomezi Feb 25 '16 at 9:09
  • $\begingroup$ Why do you think it will be equal to 1? $\endgroup$ – bulbasaur Feb 25 '16 at 9:21
  • $\begingroup$ This remark shows that it's not a good idea to think about a limit $L$ as being the (only?!) number your expression $f(h)$ gets "closer and closer" to, since that's the case for infinitely many numbers. It is however the only number where the expression $f(h)$ gets "arbitrarily close" to, i.e. "as close as you want". $\endgroup$ – StackTD Feb 25 '16 at 9:27
  • $\begingroup$ @StackTD:I couldn't get this sentence you mentioned "since that's the case for infinitely many numbers". $\endgroup$ – justin Feb 25 '16 at 9:30
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    $\begingroup$ that was badly written! It means that as h gets closer to 0. 2+h^2 gets closer to ALL the numbers less than 2. It gets closer to 1, it gets closer to zero. It gets closer to -27, etc. So the point is that isn't the Def of limit. It doesn't just get closer to the limit; the distance from the limit gets closer to 0. That was TERRIBLY written. An example of a bizarre error should never be harder to understand than the correct way of doing things. $\endgroup$ – fleablood Feb 25 '16 at 9:59
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As $h\to0$, the value of $2+h^2$ decreases, and so, gets closer to $1$. However, that does not mean that $$ \lim_{h\to0^+}\left(2+h^2\right)=1\tag{1} $$ For $\lim\limits_{h\to0^+}\left(2+h^2\right)=L$, we need that no matter how small we make $\varepsilon\gt0$, we can find a $\delta\gt0$ so that if $|h|\le\delta$, then $\left|\left(2+h^2\right)-L\right|\le\varepsilon$. This can be done with $L=2$, but not with $L=1$.

Therefore, setting $\delta=\sqrt{\varepsilon}$, we get $$ \lim_{h\to0}\left(2+h^2\right)=2\tag{2} $$

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  • $\begingroup$ :Could you tell why the value of $(2+h^2)$ decreases as $h\to0^+$? $\endgroup$ – justin Feb 25 '16 at 9:26
  • $\begingroup$ As $h\to0$, $h^2$ decreases to $0$. That is, the smaller that $|h|$ is, the smaller $2+h^2$ is. As $2+h^2$ decreases, it gets closer to $1$. It is still bounded away from $1$ by a distance of $1$, but it keeps getting closer to $1$. $\endgroup$ – robjohn Feb 25 '16 at 9:30
  • $\begingroup$ :Could you tell how could $(2+h^2)$ decrease because I can't see any subtraction or division taking place. $\endgroup$ – justin Feb 25 '16 at 9:33
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    $\begingroup$ It's horribly written. It assumes a student will make an error that no student ever has. Instead of 1, take -27. -27 < 2+ h^2 < 2+ (h+)^2. So 2 + h^2, "gets closer" to -27. But it doesn't get arbitrarily close. It is always at least 29 away. So "get closer" to isn't a good definition for limit. That's all it's saying. $\endgroup$ – fleablood Feb 25 '16 at 10:05
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    $\begingroup$ @justin: Map of California. The exact distances are not important. $\endgroup$ – robjohn Feb 25 '16 at 10:24
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This excerpt is a candidate for one of the worst written things I have ever read.

Pre-amble: In the fifties and early sixties there was a radio comedy duo called Bob and Ray and they once did a comedy skit "Wally the Word Man and Mr. Wise Old Owl on Prodigy Street" which was a satire of a children's educational television show. In it Wally the Word mad tries to teach the children the letters of the alphabet and when he gets to "I" Mr. Wise Old Owl point "notice, children, the the letter I looks like the number 1, 1 is first number when we count." "Thank you Mr. Wise Old Owl but we are teaching the children to read and although this coincidentally might look a little like the number one it is the letter I and something else entirely". When they get to the letter "O" Mr. Wise Old Owls "And, see children, the O looks like the number 0, zero actually the first number but when we count we start with the number 1 even though it is the second number." "Mr. Wise Old Owl, I really wish you wouldn't do this. I'm afraid it confuses the children." "But Wally the Word Man, I'm just trying to point this out so the children won't make the mistake." Then Wally the Word Man tries to teach them to read the word "ATTENTION" and Mr. Wise Old Owl says "And see children, the I and O look like the number 10 and the word attention says 'ten' in the middle but the the 10 in attention is in a different place than the I and the O". "Okay, that's it! You and me, Mr. Wise Old Owl, out inn the alley! ..."

I almost feel this excerpt is very similar. It's trying to point out to the reader what a limit is NOT by pointing out that $\lim_{h\rightarrow 0} 2 + h^2 \ne 1$ ("Don't make that mistake, children! The limit is not really 1!") which leaves the readers scratching their heads muttering "why the F@!# would I think the limit is $*1*$?!?!?!" So they they go to stack exchange and go through huge and confusing discussions how if you take certain misconceptions by a faulty argument one can say "2 + h^2" can "get closer to 1" to which the readers say "I still don't get it and what is so special about the number 1?" and bang their heads against the wall because they can not understand a subtelty of a badly explained !MISTAKE!.

You know something is wrong when that happens.

So this is how Uncle Fleablood would explain it:

Most students have an idea what claiming $\lim_{h\rightarrow 0} 2 + h^2 =2$ means; The values of the expression approach and hone in toward the value 2. However coming up with a formal definition for "honing in" might be more tricky. A novice might incorrectly say "the sequence will eventually get to 2" but this isn't true because the sequence itself may be infinite will never actually have any final value or 2. Many mathematicians, even professionals and professors, might say "the values of the sequence get closer to 2". This is true but this alone, as expressed, is an imprecise and insufficient to be the definition of a sequence. After all, $2 + h^2$ is a decreasing sequence and it not only "gets closer" to 2; it "gets closer" to every number that's 2 or lower! (The sequence gets closer to, say, -27 but it always stays more than 29 away-- but it gets closer and closer to being 29 away!) "Getting closer" doesn't mean "getting arbitrarily close". A precise definition of limit must not only express the idea of the sequence getting "closer" to a value, but also getting arbitrarily close to the value.

And that is ALL the author meant when s/he said $2 + h^2$ gets closer to 1. It does! It also gets closer to $0$, to $-1,$ to $-6,532*\sqrt{\pi}$ etc.


So the upshot is, the definition of limit will include not just that the sequence gets close to the value, but for any positive distance, we can find a point in the sequence where all the terms are closer to the value than the arbitrary positive distance.

$2 + h^2$ does get closer to 1, but $|(1) - (2 + h^2)| = 1 + h^2 > 1$. Always. so the sequence never gets closer than 1 to 1.

$2 + h^2$ gets closer to 2, AND $|(2) - (2 + h^2)| = h^2$ and although $h^2 > 0$ always but for any positive $\epsilon > 0$ there is always a point where $h^2 < \epsilon$ so $2 + h^2$ gets arbitrarily close to 2. But it doesn't get arbitrarily close to 1.

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  • $\begingroup$ +1 for taking the time to write this, and thanks for Bob and Ray. $\endgroup$ – Ethan Bolker Feb 26 '16 at 0:15
  • $\begingroup$ @fleablood:That's a great answer.Do you mean to say that we can take any value for $h$?If that's true we can make $2+h^2$ as a decreasing function by substituting values for $h$ as $0,-1,-2$ or any number below $0$.Am I wrong? $\endgroup$ – justin Feb 29 '16 at 5:52
  • $\begingroup$ YOu are way overthinking this. but, no, if h < 0 then $h^2 > 0$ and as h gets further from 0 (in either direction) $h^2$ gets further from 0 in either direction. $\endgroup$ – fleablood Feb 29 '16 at 16:41
  • $\begingroup$ @fleablood:Could you tell why $2+h^2$ gets arbitrarily close to $2$?I couldn't get how $2+h^2$ becomes $2$ even if we substitute negative or positive vaues for $h$. $\endgroup$ – justin Mar 2 '16 at 8:32
  • $\begingroup$ It never equals 2 if h never equals 0. But we can get it "as close to 2 as we like". If we want to get within 1/1,000,000, we can by take h $\le$ 1,000. If we want to get within 1-googolth, we can by taking h to be the square root of 1-googolth. "as close to as we like" is not very rigorous and well defined mathematically, but the term "arbitrarily close" does have a rigorous and well defined mathematical definition. .... to be continued... $\endgroup$ – fleablood Mar 2 '16 at 17:02
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$$\lim_{x\rightarrow a}{f(x)}=L$$ implies and is implied by Given any $\epsilon>0$, there exists a $\delta>0$ such that $|f(x)-L|<\epsilon$ for all $|x-a|<\delta$.

This can be rephrased as follows:

The function $f(x)$ tends to $L$ as $x$ tends to $a$ if we can make $f(x)$ as close to $L$ as we wish by making $x$ sufficiently close to $a$.

So, what is being said here is when $f(x)\rightarrow L$ when $x\rightarrow0$ it is not enough that that $f(x)$ gets closer to $L$ as $x$ gets closer to $a$, it means that by making $x$ sufficiently close to $a$, you can make $f(x)$ arbitrarily close to $L$.

This is because, say $h=100$. Then, $2+h^2=10002$. Now, you bring $h$ closer to $0$ by making $h=10$. Then, $2+h^2=102$. It can always be said that now, $h$ is coming closer to $0$ (since $|102-0|<|10002-0|$), and $2+h^2$ is coming closer to $1,2,3.6,-12.3221,...$, any number $n$ such that $|10002-n|>|102-n|$. However, the limit is $2$ as only $2$ satisfies the criteria above.

In your example, no matter how close $h$ is to $0$, $|2+h^2 - 1|>1$.

The fact that $1$ is not the limit can be verified by the $\epsilon-\delta$ test, by taking any $\epsilon<1$.

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  • $\begingroup$ Coincidently, the answer is similar to StackTD's comment. $\endgroup$ – GoodDeeds Feb 25 '16 at 9:32
  • $\begingroup$ :Could you mention what's $\epsilon$ and $\delta$? $\endgroup$ – justin Feb 25 '16 at 9:34
  • $\begingroup$ @justin $\epsilon$ (epsilon) is any arbitrary positive number. Given $\epsilon$, we need to find a positive $\delta$(delta), such that the above holds. This is the definition of a limit. $\endgroup$ – GoodDeeds Feb 25 '16 at 9:36
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    $\begingroup$ Maybe getting closer to the moon is a bad example. I wanted an extreme example. The moon is 2,159 miles in diameter. 6 feet 1 inch is closer to 2,159 miles than 5 feet 7 inches is. "closer" means "the diference between them is less". London is closer to the south pole than Liverpool is. \$5.27 is closer to the national budget than \$5.26. 27 nonometers is closer to 257 billion lightyears than 26 nanometers. etc. $\endgroup$ – fleablood Feb 29 '16 at 16:34
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    $\begingroup$ @fleablood:Yeah you're really cool and helpful.I could see this from your comments that you go upto any extreme level.That's very good. $\endgroup$ – justin Mar 2 '16 at 8:23
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Simply observe that the function $f(h)=2+h^2$ is continous on the whole $\Bbb R$, hence in particular in any nhbd of $0$, thus $$ \lim_{h\to0}f(h)=f(0) $$ and clearly $f(0)=2$.

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