Limits! math problem-not homework

becomingagodo · 12th September 2007, 01:18 PM

How do you do this
a. x^2-4x-5/x-5 where x tends to 5
and
b. x^3-125/x-5 where x tends to 5

How do you find the limit?
What is a limit?
How do you use limits?
I don't know the first question, however I am under the impression that a limit is basically near apporximation to a anwser like in the top problem I would put in 4.8 and then slowly increase it to 5. Some how it gets you the anwser or what value it will end up.

I don't know how to use a limit.
Can someone explain algebratically what is happening and how to do it.

Non standard calculus looks so much easier, their is no limit nonsense just hyper real numbers. Normal mathematician would do anything to avoid infinty, even kill a person.

rwguinn · 12th September 2007, 01:40 PM

Originally Posted by becomingagodo

How do you do this
a. x^2-4x-5/x-5 where x tends to 5
and
b. x^3-125/x-5 where x tends to 5

How do you find the limit?
What is a limit?
How do you use limits?
I don't know the first question, however I am under the impression that a limit is basically near apporximation to a anwser like in the top problem I would put in 4.8 and then slowly increase it to 5. Some how it gets you the anwser or what value it will end up.

I don't know how to use a limit.
Can someone explain algebratically what is happening and how to do it.

Non standard calculus looks so much easier, their is no limit nonsense just hyper real numbers. Normal mathematician would do anything to avoid infinty, even kill a person.

if that is $(x^2-4x-5)/(x-5)$ as I suspect,, obviously Lim(x)=5, because if x is 5, the value of the function is undefined (value/0=undefined)
Limit (x tends to 5) means that x can approach 5, but never equal it--and it can approach from either side >5 or <5

Complexity · 12th September 2007, 01:46 PM

BAGO - As you so intelligently observed (Forum Community / Asian People Superior thread ),

"No mathematician understands math"

so I'm afraid that I am unable to help you.

bjb · 12th September 2007, 01:48 PM

The easiest way to understand this is to factor the numerator to be:

(x+1)(x-5)

Then the (x-5) terms cancel out and you're left with:

lim(x+1) as x goes to 5, which is equal to 6.

There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it.

becomingagodo · 12th September 2007, 01:55 PM

Quote:

"No mathematician understands math"

so I'm afraid that I am unable to help you.

Neural networks

Quote:

if that is as I suspect,, obviously Lim(x)=5, because if x is 5, the value of the function is undefined (value/0=undefined)
Limit (x tends to 5) means that x can approach 5, but never equal it--and it can approach from either side >5 or <5

I can understand that now, however what is the anwser when x is approaching 5. How do you work that out?

Quote:

(x+1)(x-5)

Then the (x-5) terms cancel out and you're left with:

lim(x+1) as x goes to 5, which is equal to 6.

There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it.

Oh yeah, thanks.

Jimbo07 · 12th September 2007, 02:13 PM

Originally Posted by bjb

There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it.

I would, but becomingagodo just said

Quote:

Non standard calculus looks so much easier, their is no limit nonsense

So why bother?

becomingagodo · 12th September 2007, 02:18 PM

Quote:

So why bother?

Well, I want to understand the other side of calculus, you know where limits are used instead of hyper real. Plus I need to know it, so you can explain what you know or let me suffer for hours trying to get to grips with limits.
If I took the easy route then I would go nowhere, so I want to take the hard route.

Jimbo07 · 12th September 2007, 02:24 PM

Originally Posted by becomingagodo

Well, I want to understand the other side of calculus, you know where limits are used instead of hyper real.

That's comendable!

Quote:

Plus I need to know it, so you can explain what you know or let me suffer for hours trying to get to grips with limits.

Struggling with it for hours is the only way TO understand it. A calculus teacher only points you down the road... you have to travel it.

Quote:

If I took the easy route then I would go nowhere, so I want to take the hard route.

Fair enough. Be prepared to do a number of problems. This has been mentioned before. Good luck with this...

(BTW, lest you believe I don't know what I'm talking about with this example, remember that I mentioned l'Hospital's Rule when you get to it)

rwguinn · 12th September 2007, 02:29 PM

Originally Posted by bjb

The easiest way to understand this is to factor the numerator to be:

(x+1)(x-5)

Then the (x-5) terms cancel out and you're left with:

lim(x+1) as x goes to 5, which is equal to 6.

There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it.

There you go ,making things easy...
It's supposed to be confuzing!

becomingagodo · 12th September 2007, 02:34 PM

Quote:

(BTW, lest you believe I don't know what I'm talking about with this example, remember that I mentioned l'Hospital's Rule when you get to it)

So their is no easy derivative solution to getting the value of a equation when x approaches a number or infinty. I will skip functions and come back later. Mathematician

I will just stick to the approximation method.

Complexity · 12th September 2007, 03:05 PM

Suffer.

joobz · 12th September 2007, 03:17 PM

why is everyone making this difficult
enter the equation into excel
and start entering in values of 4.8, 4.9, 4.99, 4.999, 4.9999 and see where it converges.

BTW, Jimbo7's hint should be enough to get you to the easy solution.

Jimbo07 · 12th September 2007, 03:22 PM

It isn't easy, in the sense that a person should have an idea when to apply the rule, and when not to apply it.

However, if you're evaluating a difficult limit (this one is easy), it can be somewhat... hmm... convenient.

Terry · 12th September 2007, 03:56 PM

http://mathworld.wolfram.com/LHospitalsRule.html

Dylab · 12th September 2007, 04:25 PM

Just Taylor expand top and bottom. I think that is this the basic idea behind L'Hospital's rule but I think it makes things clearer.

drkitten · 13th September 2007, 08:17 AM

Originally Posted by becomingagodo

How do you find the limit?
What is a limit?
How do you use limits?

Three questions, which I will try to answer for you in a reasonably contempt-free way. However, you may still not like the answer.

What is a limit? A limit is a recognition that while a particular function may not have a mathematically tractable answer at a specific point, it is nevertheless to get arbitrarily close to that point -- and that when you do, you get arbitrarily close to another value that is the "limit" of the function at that point.

In particular:
$\frac{x^{2} -4x-5}{x-5}$ = $\frac{(x-5)(x+1)}{x-5}$ equals $\frac{0}{0}$ at x=5. Since 0/0 doesn't have a value. there is a "hole" in the function there. But if you graph the function, you will see that it is a straight line (y = x + 1) everywhere else. So plotting 4.9, 4.99, 4.999 and so forth gets you 5.9, 5.99, 5.999 and so on. If you set x close enough to 5, y will be very close to 6.

Now, that's the full-on hard-core mathematical definition. MOST of the time you don't need to use anything that rigorous to find a limit.

How do you find the limit? Well, if you're really in trouble, you can go through the successive-approximation route. Most of the time, however, the problems that you're interested in solving have a structure that allows for certain "tricks" to be applied (like the graphing trick I showed, or L'Hopital's rule). (It's like your Cramer's Rule question earlier. The more techniques you know, the better your chances of knowing a technique that is faster/cheaper/more effective that the most general solution.)

How do you use limits? The most common "use" of limits is as a teaching technique so you understand what is really meant by various calculus concepts like the derivative. The derivative, in particular, is another example of dividing zero by zero (since it's an instantaneous rate of change, the elapsed time is zero and the elapsed change is also zero). How can we avoid this? Simply define the derivative as the limit of change over successively smaller intervals.

Now, in most cases, you won't need to take the limit yourself to calculate the derivative, because there will be tricks available (depending upon the type of function you're dealing with.) But you need to understand the limit concepts to know what the tricks do and how to use them.

Jimbo07 · 13th September 2007, 12:26 PM

Originally Posted by drkitten

allows for certain "tricks" to be applied (like the graphing trick I showed, or L'Hopital's rule). (It's like your Cramer's Rule question earlier. The more techniques you know, the better your chances of knowing a technique that is faster/cheaper/more effective that the most general solution.)

I once thought that math was a form of magic that I would never understand. It quite literally reduced me to tears some evenings, but...

Like any skill, people develop a 'toolbox' of techniques. I know that there are higher maths that I don't know, but I'd bet anything that daily practitioners also have a toolbox of techniques.

This thread is a great example of the value of a common language! Instead of science and math limiting the imagination, instead, everyone in this thread is able to understand what each other is talking about, given a certain educational level. Math facilitates communication... except for certain uses of LaTeX.

Once a person has a certain understanding of a subject, rather than the mind being closed, the mind is freed to move beyond the basics and tackle new and interesting problems!

drkitten · 13th September 2007, 02:03 PM

Originally Posted by Jimbo07

Like any skill, people develop a 'toolbox' of techniques. I know that there are higher maths that I don't know, but I'd bet anything that daily practitioners also have a toolbox of techniques.

Yes, but don't overrely on the "toolbox of techniques." More important than understanding techniques is understanding the problem, because there's almost always a simple, slow way of solving any mathematical problem that is guaranteed to work, and then a set of tricks that will often work in "normal" circumstances. If you don't need to solve problems all that often, and you have a little patience, you can almost always work stuff out the long way, and you still get the right answer.

Counting on your fingers, for example, is a fine way of doing mathematics if you're five years old. And it's better to count on your fingers and get the answer right than it is to get the answer wrong because you haven't managed to memorize the tables yet.

Part of the problem that many people have with math is that they see the fluency with which experts can pull stuff off, and they don't realize that that very fluency is a mark of the expertise.

12th September 2007, 01:18 PM	#1
becomingagodo Banned Join Date: Mar 2007 Posts: 698	Limits! math problem-not homework How do you do this a. x^2-4x-5/x-5 where x tends to 5 and b. x^3-125/x-5 where x tends to 5 How do you find the limit? What is a limit? How do you use limits? I don't know the first question, however I am under the impression that a limit is basically near apporximation to a anwser like in the top problem I would put in 4.8 and then slowly increase it to 5. Some how it gets you the anwser or what value it will end up. I don't know how to use a limit. Can someone explain algebratically what is happening and how to do it. Non standard calculus looks so much easier, their is no limit nonsense just hyper real numbers. Normal mathematician would do anything to avoid infinty, even kill a person.
becomingagodo Banned Join Date: Mar 2007 Posts: 698	Last edited by becomingagodo; 12th September 2007 at 01:23 PM.

12th September 2007, 01:40 PM	#2
rwguinn Philosopher Join Date: Apr 2003 Location: 16 miles from 7 lakes Posts: 8,432	Originally Posted by becomingagodo How do you do this a. x^2-4x-5/x-5 where x tends to 5 and b. x^3-125/x-5 where x tends to 5 How do you find the limit? What is a limit? How do you use limits? I don't know the first question, however I am under the impression that a limit is basically near apporximation to a anwser like in the top problem I would put in 4.8 and then slowly increase it to 5. Some how it gets you the anwser or what value it will end up. I don't know how to use a limit. Can someone explain algebratically what is happening and how to do it. Non standard calculus looks so much easier, their is no limit nonsense just hyper real numbers. Normal mathematician would do anything to avoid infinty, even kill a person. if that is $(x^2-4x-5)/(x-5)$ as I suspect,, obviously Lim(x)=5, because if x is 5, the value of the function is undefined (value/0=undefined) Limit (x tends to 5) means that x can approach 5, but never equal it--and it can approach from either side >5 or <5
	__________________ "Political correctness is a doctrine,...,which holds forth the proposition that it is entirely possible to pick up a turd by the clean end." "I pointed out that his argument was wrong in every particular, but he rightfully took me to task for attacking only the weak points." Myriad http://forums.randi.org/showthread.php?postid=6853275#post6853275

12th September 2007, 01:46 PM	#3
Complexity The Woo Whisperer Join Date: Nov 2005 Location: Minnesota, USA Posts: 9,263	BAGO - As you so intelligently observed (Forum Community / Asian People Superior thread ), "No mathematician understands math" so I'm afraid that I am unable to help you.
	__________________ "It is a great nuisance that knowledge can only be acquired by hard work." - W. Somerset Maugham "Thought is subversive and revolutionary, destructive and terrible; thought is merciless to privilege, established intuititions, and comfortable habit. Thought looks into the pit of hell and is not afraid. Thought is great and swift and free, the light of the world, and the chief glory of man." - Bertrand Russell

12th September 2007, 01:48 PM	#4
bjb Graduate Poster Join Date: Jan 2005 Posts: 1,079	The easiest way to understand this is to factor the numerator to be: (x+1)(x-5) Then the (x-5) terms cancel out and you're left with: lim(x+1) as x goes to 5, which is equal to 6. There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it.
bjb Graduate Poster Join Date: Jan 2005 Posts: 1,079

12th September 2007, 01:55 PM	#5
becomingagodo Banned Join Date: Mar 2007 Posts: 698	Quote: "No mathematician understands math" so I'm afraid that I am unable to help you. Neural networks Quote: if that is as I suspect,, obviously Lim(x)=5, because if x is 5, the value of the function is undefined (value/0=undefined) Limit (x tends to 5) means that x can approach 5, but never equal it--and it can approach from either side >5 or <5 I can understand that now, however what is the anwser when x is approaching 5. How do you work that out? Quote: (x+1)(x-5) Then the (x-5) terms cancel out and you're left with: lim(x+1) as x goes to 5, which is equal to 6. There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it. Oh yeah, thanks.
becomingagodo Banned Join Date: Mar 2007 Posts: 698	Last edited by becomingagodo; 12th September 2007 at 01:58 PM.

12th September 2007, 02:13 PM	#6
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	Originally Posted by bjb There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it. I would, but becomingagodo just said Quote: Non standard calculus looks so much easier, their is no limit nonsense So why bother?
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	__________________ This post approved by your local jPac (Jimbo07 Political Action Committee), also registered with Jimbo07 as the Jimbo07 Equality Rights Knowledge Betterment Action Group. Atoms in supernova explosion get huge business -- Pixie of key

12th September 2007, 02:18 PM	#7
becomingagodo Banned Join Date: Mar 2007 Posts: 698	Quote: So why bother? Well, I want to understand the other side of calculus, you know where limits are used instead of hyper real. Plus I need to know it, so you can explain what you know or let me suffer for hours trying to get to grips with limits. If I took the easy route then I would go nowhere, so I want to take the hard route.
becomingagodo Banned Join Date: Mar 2007 Posts: 698

12th September 2007, 02:24 PM	#8
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	Originally Posted by becomingagodo Well, I want to understand the other side of calculus, you know where limits are used instead of hyper real. That's comendable! Quote: Plus I need to know it, so you can explain what you know or let me suffer for hours trying to get to grips with limits. Struggling with it for hours is the only way TO understand it. A calculus teacher only points you down the road... you have to travel it. Quote: If I took the easy route then I would go nowhere, so I want to take the hard route. Fair enough. Be prepared to do a number of problems. This has been mentioned before. Good luck with this... (BTW, lest you believe I don't know what I'm talking about with this example, remember that I mentioned l'Hospital's Rule when you get to it)
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	__________________ This post approved by your local jPac (Jimbo07 Political Action Committee), also registered with Jimbo07 as the Jimbo07 Equality Rights Knowledge Betterment Action Group. Atoms in supernova explosion get huge business -- Pixie of key

12th September 2007, 02:29 PM	#9
rwguinn Philosopher Join Date: Apr 2003 Location: 16 miles from 7 lakes Posts: 8,432	Originally Posted by bjb The easiest way to understand this is to factor the numerator to be: (x+1)(x-5) Then the (x-5) terms cancel out and you're left with: lim(x+1) as x goes to 5, which is equal to 6. There's another, faster way to do this using derivatives, but I'll give someone else a chance to explain it. There you go ,making things easy... It's supposed to be confuzing!
	__________________ "Political correctness is a doctrine,...,which holds forth the proposition that it is entirely possible to pick up a turd by the clean end." "I pointed out that his argument was wrong in every particular, but he rightfully took me to task for attacking only the weak points." Myriad http://forums.randi.org/showthread.php?postid=6853275#post6853275

12th September 2007, 02:34 PM	#10
becomingagodo Banned Join Date: Mar 2007 Posts: 698	Quote: (BTW, lest you believe I don't know what I'm talking about with this example, remember that I mentioned l'Hospital's Rule when you get to it) So their is no easy derivative solution to getting the value of a equation when x approaches a number or infinty. I will skip functions and come back later. Mathematician I will just stick to the approximation method.
becomingagodo Banned Join Date: Mar 2007 Posts: 698	Last edited by becomingagodo; 12th September 2007 at 02:36 PM.

12th September 2007, 03:05 PM	#11
Complexity The Woo Whisperer Join Date: Nov 2005 Location: Minnesota, USA Posts: 9,263	Suffer.
	__________________ "It is a great nuisance that knowledge can only be acquired by hard work." - W. Somerset Maugham "Thought is subversive and revolutionary, destructive and terrible; thought is merciless to privilege, established intuititions, and comfortable habit. Thought looks into the pit of hell and is not afraid. Thought is great and swift and free, the light of the world, and the chief glory of man." - Bertrand Russell

12th September 2007, 03:17 PM	#12
joobz Tergiversator Join Date: Aug 2006 Location: That's how you get ants Posts: 17,496	why is everyone making this difficult enter the equation into excel and start entering in values of 4.8, 4.9, 4.99, 4.999, 4.9999 and see where it converges. BTW, Jimbo7's hint should be enough to get you to the easy solution.
	__________________ What's the best argument for UHC? This argument against UHC. "Perhaps one reason per capita GDP is lower in UHC countries is because they've tried to prevent this important function [bankrupting the sick] and thus carry forward considerable economic dead wood?"-BeAChooser

12th September 2007, 03:22 PM	#13
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	It isn't easy, in the sense that a person should have an idea when to apply the rule, and when not to apply it. However, if you're evaluating a difficult limit (this one is easy), it can be somewhat... hmm... convenient.
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	__________________ This post approved by your local jPac (Jimbo07 Political Action Committee), also registered with Jimbo07 as the Jimbo07 Equality Rights Knowledge Betterment Action Group. Atoms in supernova explosion get huge business -- Pixie of key

12th September 2007, 03:56 PM	#14
Terry Zeitgeist-impaired Technical Moderator Join Date: Jun 2003 Location: logged in to the server Posts: 6,450	http://mathworld.wolfram.com/LHospitalsRule.html

12th September 2007, 04:25 PM	#15
Dylab Critical Thinker Join Date: Nov 2002 Posts: 313	Just Taylor expand top and bottom. I think that is this the basic idea behind L'Hospital's rule but I think it makes things clearer.
Dylab Critical Thinker Join Date: Nov 2002 Posts: 313

13th September 2007, 08:17 AM	#16
drkitten Penultimate Amazing Join Date: Mar 2004 Location: Wits' End Posts: 21,647	Originally Posted by becomingagodo How do you find the limit? What is a limit? How do you use limits? Three questions, which I will try to answer for you in a reasonably contempt-free way. However, you may still not like the answer. What is a limit? A limit is a recognition that while a particular function may not have a mathematically tractable answer at a specific point, it is nevertheless to get arbitrarily close to that point -- and that when you do, you get arbitrarily close to another value that is the "limit" of the function at that point. In particular: $\frac{x^{2} -4x-5}{x-5}$ = $\frac{(x-5)(x+1)}{x-5}$ equals $\frac{0}{0}$ at x=5. Since 0/0 doesn't have a value. there is a "hole" in the function there. But if you graph the function, you will see that it is a straight line (y = x + 1) everywhere else. So plotting 4.9, 4.99, 4.999 and so forth gets you 5.9, 5.99, 5.999 and so on. If you set x close enough to 5, y will be very close to 6. Now, that's the full-on hard-core mathematical definition. MOST of the time you don't need to use anything that rigorous to find a limit. How do you find the limit? Well, if you're really in trouble, you can go through the successive-approximation route. Most of the time, however, the problems that you're interested in solving have a structure that allows for certain "tricks" to be applied (like the graphing trick I showed, or L'Hopital's rule). (It's like your Cramer's Rule question earlier. The more techniques you know, the better your chances of knowing a technique that is faster/cheaper/more effective that the most general solution.) How do you use limits? The most common "use" of limits is as a teaching technique so you understand what is really meant by various calculus concepts like the derivative. The derivative, in particular, is another example of dividing zero by zero (since it's an instantaneous rate of change, the elapsed time is zero and the elapsed change is also zero). How can we avoid this? Simply define the derivative as the limit of change over successively smaller intervals. Now, in most cases, you won't need to take the limit yourself to calculate the derivative, because there will be tricks available (depending upon the type of function you're dealing with.) But you need to understand the limit concepts to know what the tricks do and how to use them.
	Last edited by drkitten; 13th September 2007 at 08:20 AM.

13th September 2007, 12:26 PM	#17
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	Originally Posted by drkitten allows for certain "tricks" to be applied (like the graphing trick I showed, or L'Hopital's rule). (It's like your Cramer's Rule question earlier. The more techniques you know, the better your chances of knowing a technique that is faster/cheaper/more effective that the most general solution.) I once thought that math was a form of magic that I would never understand. It quite literally reduced me to tears some evenings, but... Like any skill, people develop a 'toolbox' of techniques. I know that there are higher maths that I don't know, but I'd bet anything that daily practitioners also have a toolbox of techniques. This thread is a great example of the value of a common language! Instead of science and math limiting the imagination, instead, everyone in this thread is able to understand what each other is talking about, given a certain educational level. Math facilitates communication... except for certain uses of LaTeX. Once a person has a certain understanding of a subject, rather than the mind being closed, the mind is freed to move beyond the basics and tackle new and interesting problems!
Jimbo07 Illuminator Join Date: Jan 2006 Posts: 3,417	__________________ This post approved by your local jPac (Jimbo07 Political Action Committee), also registered with Jimbo07 as the Jimbo07 Equality Rights Knowledge Betterment Action Group. Atoms in supernova explosion get huge business -- Pixie of key

13th September 2007, 02:03 PM	#18
drkitten Penultimate Amazing Join Date: Mar 2004 Location: Wits' End Posts: 21,647	Originally Posted by Jimbo07 Like any skill, people develop a 'toolbox' of techniques. I know that there are higher maths that I don't know, but I'd bet anything that daily practitioners also have a toolbox of techniques. Yes, but don't overrely on the "toolbox of techniques." More important than understanding techniques is understanding the problem, because there's almost always a simple, slow way of solving any mathematical problem that is guaranteed to work, and then a set of tricks that will often work in "normal" circumstances. If you don't need to solve problems all that often, and you have a little patience, you can almost always work stuff out the long way, and you still get the right answer. Counting on your fingers, for example, is a fine way of doing mathematics if you're five years old. And it's better to count on your fingers and get the answer right than it is to get the answer wrong because you haven't managed to memorize the tables yet. Part of the problem that many people have with math is that they see the fluency with which experts can pull stuff off, and they don't realize that that very fluency is a mark of the expertise.