My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree6Thanks
Reply
 
LinkBack Thread Tools Display Modes
December 12th, 2016, 05:55 PM   #1
Newbie
 
Joined: Dec 2016
From: Sweden

Posts: 13
Thanks: 1

Hard limit problem one variable calculus

Made errors in earlier posts. This is my question.

I have been trying to solve this since summer.

It is a question from the book: 7th edition Adams calculus.
2.1.3

Using Newton quotient to solve the problem (not any chain rule, power rule or l'Hôpital's rule
is allowed).
If the function $f$ has a tangent at the given point? If yes, what is the tangent line?

$\displaystyle f(x)=\sqrt{|x|}\: \text{at} \: x=0$

It looks obvious at first:

$\displaystyle \lim_{h\to 0}\frac{\sqrt{|0+h|}-\sqrt{0}}{h}=$

$\displaystyle =\:\lim_{h\to 0}\frac{\sqrt{|h|}}{h}$

Which does seem right. I can draw the needed conclusion I think?
It is a two-sided limit, which does not exist in this case.

But when I look in the book. The solutions manual.

It states the same conclusion as me, but with a different way of reaching it.

It says:

$\displaystyle \lim_{h\to 0}\frac{\sqrt{|0+h|}-\sqrt{0}}{h}=$

Up until this point I understand, but then the solutions manual Bends My mind Into deep insanity with the above claiming to be equal to

$\displaystyle =\lim_{h\to 0}\frac{1}{|h|\text{sgn}(h)}$

Can someone please explain to me how this last step is achieved?

Is it just a logical step I am supposed to make to show an equal situation as above? Where the limit goes to negative and positive infinity?

Or is there really a way to rewrite

$\displaystyle \lim_{h\to 0}\frac{\sqrt{|0+h|}-\sqrt{0}}{h}=$

into

$\displaystyle \lim_{h\to 0}\frac{1}{|h|\text{sgn}(h)}$

??????

I would really love an answer.

Thank you.

This last step is taken from The instructor's solutions manual.

I am assuming it is completely correct.

Last edited by skipjack; December 13th, 2016 at 06:07 AM.
Luciferis is offline  
 
December 13th, 2016, 01:44 AM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 19,542
Thanks: 1751

The function $f(x)$ is defined, but not differentiable, at $x = 0$. Its graph has a tangent (with equation $x = 0$) at $x = 0$.

Your use of the difference quotient in your post is correct. The solutions manual, if you have quoted it accurately, contains a mistake (a typographical error or slip by the author).
skipjack is offline  
December 13th, 2016, 03:20 AM   #3
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,261
Thanks: 894

It does not help you to post the same question repeatedly!
Country Boy is offline  
December 13th, 2016, 04:55 AM   #4
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,403
Thanks: 2477

Math Focus: Mainly analysis and algebra
1) Your book is wrong.
2) $\DeclareMathOperator{\sgn}{sgn} h=|h| \sgn{(x)}$ by definition.
3) The tangent would be the $y$ axis.

Last edited by v8archie; December 13th, 2016 at 04:58 AM.
v8archie is offline  
December 13th, 2016, 06:09 AM   #5
Global Moderator
 
Joined: Dec 2006

Posts: 19,542
Thanks: 1751

I take it you mean the $f(x)$ axis, as there is no $y$ in this problem.

I've merged the other thread into this one.
skipjack is offline  
December 13th, 2016, 01:48 PM   #6
Newbie
 
Joined: Dec 2016
From: Sweden

Posts: 13
Thanks: 1

Quote:
Originally Posted by skipjack View Post
The function $f(x)$ is defined, but not differentiable, at $x = 0$. Its graph has a tangent (with equation $x = 0$) at $x = 0$.

Your use of the difference quotient in your post is correct. The solutions manual, if you have quoted it accurately, contains a mistake (a typographical error or slip by the author).
IMG_0909.jpg

Certain?
It is not some type of text proof like situation where something is similar to the real answer and I should understand it?

That I should use this |h|sgn(h) as a replacement for a situation where
things look the same close to 0?

It made me feel stupid.

An answer to this would be helpful.
I have been retaking calclulus for years. And this is the only one on the first half of the cource I cannot solve. Except for epsilon delta proofs.
Luciferis is offline  
December 13th, 2016, 02:53 PM   #7
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 444
Thanks: 254

Math Focus: Dynamical systems, analytic function theory, numerics
Maybe the comments above relate to the other thread but the quote you have from the book is not wrong. It is exactly correct. For $h > 0$ you have $f(h) = \sqrt{h}$ and therefore
\[
\lim_{h \rightarrow 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h \rightarrow 0^+} \frac{1}{\sqrt{h}} = \infty.
\]
Note that if $h < 0$ you have $f(h) = \sqrt{-h}$ and a similar computation gives you a limit of $-\infty$. This should make sense graphically since the slopes of $f(x)$ increase without bound as $x$ tends to 0 from the right and they decrease without bound from the left. Refer to the graph to see this geometrically.
Wolfram|Alpha: Computational Knowledge Engine

It is also not true that there is a vertical tangent line at the $y$-axis. In fact there is a cusp here and tangency loses meaning.
SDK is offline  
December 13th, 2016, 03:24 PM   #8
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,403
Thanks: 2477

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by SDK View Post
\[
\lim_{h \rightarrow 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h \rightarrow 0^+} \frac{1}{\sqrt{h}} \]
That's not what the book says though. It would probably be something like $\displaystyle \lim_{h \rightarrow 0} \frac{f(0+h) - f(0)}{h} =\lim_{h \rightarrow 0}\frac{\sqrt{|h|}}{2h}$ anyway.

Quote:
Originally Posted by SDK View Post
there is [no] vertical tangent line at the $y$-axis. In fact there is a cusp here and tangency loses meaning.
This is true (hence the "would") in my post. However, if you were to define a tangent at $x=0$ it would be the $y$-axis.
Thanks from SDK

Last edited by skipjack; December 13th, 2016 at 06:05 PM.
v8archie is offline  
December 13th, 2016, 04:45 PM   #9
Newbie
 
Joined: Dec 2016
From: Sweden

Posts: 13
Thanks: 1

My problem is still with understanding
The rewriting of; $\displaystyle \lim_{h\to 0}\frac{\sqrt{|0+h|}-\sqrt{0}}{h}=$

into

$\displaystyle \lim_{h\to 0}\frac{1}{|h|\text{sgn}(h)}$

In one variable calculus. It has no tangent.
I know some measurement theory.
Did a research level crash course one summer; measurement theory.
There was some mentioning of pointy peaks being differentiable.

But no! It has no tangent!
There are other situations where an infinite limit could imply this.
But this is not it.

Please help me understand the rewriting of this.

Last edited by skipjack; December 13th, 2016 at 06:04 PM.
Luciferis is offline  
December 13th, 2016, 05:44 PM   #10
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,403
Thanks: 2477

Math Focus: Mainly analysis and algebra
I can't see any analytic way of writing the limit as they have done without first accepting that the limit is $\pm \infty$ and then recognising that any positive power of $h$ gives the same result (i.e unbounded - the direction may be different).

The decision to write $\DeclareMathOperator{\sgn}{sgn}|x| \sgn{(x)}$ instead of $x$ is frankly bizarre.

Last edited by v8archie; December 13th, 2016 at 05:47 PM.
v8archie is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
abs, calculus, hard, limit, newton, point, problem, quotient, variable



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Find the Limit Calculus Problem mathkid Calculus 2 September 16th, 2012 02:31 PM
Calculus limit problem (pic) claudillama Calculus 3 January 31st, 2012 07:52 PM
Calculus Limit Problem netmaxweb Calculus 2 October 15th, 2011 01:11 PM
Two variable calculus-1 Zeefinity Calculus 2 February 25th, 2011 06:17 AM
Two variable calculus-2 Zeefinity Calculus 2 February 15th, 2011 08:01 PM





Copyright © 2018 My Math Forum. All rights reserved.