Limits! math problem-not homework

[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.

[rwguinn]

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]

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]

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]

"No mathematician understands math"

so I'm afraid that I am unable to help you.
Neural networks:cool:

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?
(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]

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


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

So why bother?

[becomingagodo]

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]

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

That's comendable!


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.


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]

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!:D

[becomingagodo]

(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:rolleyes:

I will just stick to the approximation method.

[Complexity]

Suffer.

[joobz]

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]

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]

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

[Dylab]

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]

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]

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]

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.