L'Hopital's Rule

[Darya]

Find lim ((ln6x)^2)/(ln2x)^2 as x approaches infinity. So we get infinity over infinity but applying L'Hopitals Rule I get the same over and over again. The answer should be 9. Can you spot the mistake?

 

[Li Batton]

Remember the chain rule.

[MATH]\frac{d}{dx}(ln6x)^2=2(ln6x)(\frac{1}{6x})(6)[/MATH]

 

[Deleted member 4993]

Find lim ((ln6x)^2)/(ln2x)^2 as x approaches infinity. So we get infinity over infinity but applying L'Hopitals Rule I get the same over and over again. The answer should be 9. Can you spot the mistake?

View attachment 17701

According to my calculations, you do not need to use L'Hopital here.

[ln(6x)/ln(2x)]² = [{ln(3) + ln(2x)}/ln(2x)]² = [1 + ln(3)/ln(2x)]²

However, I am getting a limit of 1.

I did a spreadsheet calculation and saw the limit approaching 1 - very slowly.

At x =500000, I get

[ln(6x)/ln(2x)]² = 1.165363882

At x =1000000, I get

[ln(6x)/ln(2x)]² = 1.157175998

slowly decreasing and nowhere near 9!!

 

[Dr.Peterson]

First, what you claim to be doing (first and second derivatives of y) is neither what L'Hopital calls for, nor what you did! The rule is to differentiate the numerator and the denominator separately, which you did.

But why did you rearrange the expression, complicating it more, before doing that? Just differentiate the numerator and the denominator of the original expression! You will get the correct limit without much trouble; but it is 1, not 9.

(I'm assuming you are required to use L'Hopital, since it is not the only way.)

 

[Darya]

First, what you claim to be doing (first and second derivatives of y) is neither what L'Hopital calls for, nor what you did! The rule is to differentiate the numerator and the denominator separately, which you did.

But why did you rearrange the expression, complicating it more, before doing that? Just differentiate the numerator and the denominator of the original expression! You will get the correct limit without much trouble; but it is 1, not 9.

(I'm assuming you are required to use L'Hopital, since it is not the only way.)

Thanks! Now I realize that my notation is way off and that I shouldnt've made it so complicated.

 

[Darya]

According to my calculations, you do not need to use L'Hopital here.

[ln(6x)/ln(2x)]² = [{ln(3) + ln(2x)}/ln(2x)]² = [1 + ln(3)/ln(2x)]²

However, I am getting a limit of 1.

I did a spreadsheet calculation and saw the limit approaching 1 - very slowly.

At x =500000, I get

[ln(6x)/ln(2x)]² = 1.165363882

At x =1000000, I get

[ln(6x)/ln(2x)]² = 1.157175998

slowly decreasing and nowhere near 9!!

There's got to be the mistake in the keys then... In WolframAlpha, which I don't like to use tho, I also got 1. Thank you for your time and effort spent to help me!!

 

[Dr.Peterson]

This is among the best times to use WA, namely to check whether a provided answer is wrong! Like you, I would generally hold off on using it until either I have something to check, or I'm working on a problem for which I don't need the practice and just want an answer.

I assume you've checked that you copied the problem correctly, and that you looked at the answer to the right problem ...

 

[Steven G]

Recall that A²/B²= (A/B)².

I would have computed the lim for (A/B) and then squared the result simplifying the problem significantly.

 

[Darya]

Recall that A²/B²= (A/B)².

I would have computed the lim for (A/B) and then squared the result simplifying the problem significantly.

But not me, I LOVE to complicate stuff)
thanks)

 

[Steven G]

But not me, I LOVE to complicate stuff)
thanks)

Have you heard of Algebraic Topology? That stuff will make you smile all day long and you won't have to make it complicated as it always is!

 

[Darya]

Have you heard of Algebraic Topology? That stuff will make you smile all day long and you won't have to make it complicated as it always is!

I've heard of it but only that 'A doughnut is the same as a cup in that thing'
And since I've started learning maths in a normal way (mainly thanks to you guys), which doesn't include only learning formulas by heart, not a long time ago, I have a lot of stuff yet to learn. But I already love it from your description! I'll make sure I check it out))

 

[Steven G]

I've heard of it but only that 'A doughnut is the same as a cup in that thing'
And since I've started learning maths in a normal way (mainly thanks to you guys), which doesn't include only learning formulas by heart, not a long time ago, I have a lot of stuff yet to learn. But I already love it from your description! I'll make sure I check it out))

Actually what you mentioned is just topology, algebraic topology is way for fun then just plain ole topology. You'll be talking to yourself on the street corner after just a week of studying this material.

 

[Darya]

Actually what you mentioned is just topology, algebraic topology is way for fun then just plain ole topology. You'll be talking to yourself on the street corner after just a week of studying this material.

looks like something that makes you immortal after you finish the course

 

[topsquark]

Actually what you mentioned is just topology, algebraic topology is way for fun then just plain ole topology. You'll be talking to yourself on the street corner after just a week of studying this material.

Heck, when I go out to study I usually sit at the bar in a restaurant and mumble at my calculations. I've started being preemptive and tell anyone sitting next to me that, yes, I did take my medications today!

-Dan

 

[Deleted member 4993]

Actually what you mentioned is just topology, algebraic topology is way for fun then just plain ole topology. You'll be talking to yourself on the street corner after just a week of studying this material.

When the blue-tooth speakers first came out - I felt the number of "algebraic topologists" suddenly jump. I found whole bunch of people walking on the side-walk laughing, arguing, with "no-body" there!!