• We’re currently investigating an issue related to the forum theme and styling that is impacting page layout and visual formatting. The problem has been identified, and we are actively working on a resolution. There is no impact to user data or functionality, this is strictly a front-end display issue. We’ll post an update once the fix has been deployed. Thanks for your patience while we get this sorted.

Math question

P

pinion9

Banned
Essentially I had a HW problem where I needed to prove orders of growth. The way to do this is
Lim n-> infinity f(n)/g(n)
I contend that this equals
Lim n-> infinity log[f(n)]/log[g(n)].
Taking the log of certain functions makes the calculations much easier.He says that isn't true. Can someone confirm or dispel this?


 
OK, let f(n)=3 and g(n)=4.
Then lim n-> inf f(n)/g(n) = 3/4 = 0.75,
but lim n-> inf log[f(n)]/log[g(n)] = log(3)/log(4) = .79248
So your prof is right.
 
even better for you...

consider lim n -> inf of f(n)/g(n)

where f(n) = 1, and g(n) = 1

there for, lim n -> inf of f(n)/g(n) = 1

now, consider your argument.

log(f(n)) = 0, log(g(n)) = 0

lim n-> of log(f(n))/log(g(n)) = 0/0 UNDEFINED.

your whole argument falls apart for clearly defined functions f and g, let alone loosely defined functions f and g.
furthermore, log(x) is NOT defined for all values of x. it is only defined for values of x > 0.

you MIGHT have meant, lim blah blah blah f/g is related to log (f/g), but that argument also falls apart taken into account the above.

i will assume you had this lesson in the past with respect to "range" and "domains" of functions from elementary algebra. e.g. you cant get there from here.

taking logs of functions DOES NOT make things easier. *cough* chain rule *cough*
 
I see the fallacy now...very simple indeed. Thanks for being helpful, djhuber.
Mday, all I can say is *cough* don't be such an a-hole *cough* Yes, I had the lessons about that, but is has been some time since I've actually taken a decent math course and occasionally I forget these things...
 
Originally posted by: pinion9
I see the fallacy now...very simple indeed. Thanks for being helpful, djhuber.
Mday, all I can say is *cough* don't be such an a-hole *cough* Yes, I had the lessons about that, but is has been some time since I've actually taken a decent math course and occasionally I forget these things...
Click to expand...

same, you have no idea how many times i rewrote my reply before pressing post lol.
 
There are times when log (or rather, the easier natural log) is used in finding limits...

Find lim x->0 of (1 + x)^(1/x)

Let L = (1+x)^(1/x)
Then ln(L) = ln[(1+x)^(1/x)]
lim x-> 0 of ln(L) = lim x->0 of ln[(1+x)^(1/x)]
using properties of logs,
lim x-> 0 of ln(L) = lim x->0 of (1/x)*ln(1+x)
If you write the right hand side as the fraction

lim x ->0 of [ln(1+x)]/x
you have an indeterminate form of type 0/0
Applying L'Hopital's rule,
lim x->0 of ln(L) = lim x ->0 of [ln(1+x)]/x = lim x-> 0 of [1/(1+x)]/1
evaluate the limit after applying L'Hopital's rule once and you have = 1

This shows that ln(L) -> 1 as x -> 0.
Since it's a continuous function at 1,
e^ln(L) -> e^1 as x->0
This implies
L -> e as x -> 0
So, lim x->0 (1 + x)^(1/x) = e



 
He didn't do what you did... log (lim) = log (function)
He did: lim = log(function part 1)/log(function part 2).
A mistake ALL of us are guilty of in some for or another, even if we're not taking logs.
 
Originally posted by: Mday
He didn't do what you did... log (lim) = log (function)
He did: lim = log(function part 1)/log(function part 2).
A mistake ALL of us are guilty of in some for or another, even if we're not taking logs.
Click to expand...

I know what he did. I was merely pointing out that there ARE cases where logs are used when taking limits; and I showed the correct way to do so (for one example). Rather than just say "You did the problem wrong. But, there are times that logs are used when finding limits," I felt that my post would be a little more informative for the OP; he already knew he did the problem wrong. Plus he might have some recollection of what he was attempting to do, i.e. "oh yeah, that's what I was thinking of"

And, if the mistake you're referring to is thinking that a rational expression has the same value after performing some function on the numerator and denominator separately (other than multiplication or division, and maybe a couple others that I'm not going to take the time to think of right now), then it's not a mistake that I make. Just because a mistake is a common mistake doesn't mean that all of us, particularly in this forum, make the same mistake on occasion.

edit: you could add or subtract 0 to the numerator and denominator separately and it will still have the same value.

 
Originally posted by: DrPizza
And, if the mistake you're referring to is thinking that a rational expression has the same value after performing some function on the numerator and denominator separately (other than multiplication or division, and maybe a couple others that I'm not going to take the time to think of right now), then it's not a mistake that I make. Just because a mistake is a common mistake doesn't mean that all of us, particularly in this forum, make the same mistake on occasion.
Click to expand...

I call shens...

We have all made that mistake, either when we were young, half asleep, or drunk.
 
Originally posted by: Mday
We have all made that mistake, either when we were young, half asleep, or drunk.
Click to expand...

Well, if by "we", you're including yourself and a bunch of people who have only average ability in mathematics, then you may be correct. Don't include me in your "we." And for every year I've taught, I can name at least a dozen students who have and probably never will make such a mistake.

For what it's worth, a VERY common mistake is (x+y)^2 = x^2 + y^2
In fact, that mistake is among my top 10 pet peeves for students who have made it to pre-calculus or beyond. But not everyone has made that mistake at some time.
As I teach mathematics to students from algebra through calculus in a small school (which means I see some of these students multiple years) I know that I could not state that ALL students make that mistake from time to time. Because, some students master math skills so quickly that making such an error for them would be akin to misspelling "dog" as "dgo". Some people are just that good. They don't make those types of mistakes, ever. Ironically, most of them tend to make arithmetic mistakes fairly often; or perhaps more likely is that those mistakes stand out because they are the only mistakes these students make. And, the most likely reason they make arithmetic mistakes as that they are not so mentally lazy that they rely on a calculator to do arithmetic. I don't know how advanced you are in mathematics, but while majoring in mathematics, one of my final exams was a take home exam. (NEVER agree to take a take home exam.) My work totaled over 50 pages... for 4 problems. Now, if I, or any of my classmates were making the types of mistakes that you think "we all" do from time to time, none of us would have ever completed such an exam.
 
i was referring to a miss-application of function notation. applying equivalence when it should not be. whether or not they are corrected immediately does not mean the mistake was not made.
 
You must log in or register to reply here.

TRENDING THREADS

Back
Top