Limits question

[cursive edge]

I found this question on an instagram post and I want to learn how to solve it

[long marten]

Wow

Well first thing would be to give an expression to log base sin^2(x)

Log in base a=ln(x)/ln(a)

With ln(x) natural logarithm

So log in base sin^2(x)= ln(x)/ln(sin^2(x))

We can simplify it:

log_sin^2(x)=ln(x)/2ln(sin(x))

[cursive edge]

Would it be a similar simplification for the log_sin²(x/2)?

[long marten]

Also: log_sin^2(x/2)=ln(x)/ln(sin^2(x/2))

Yes

[cursive edge]

So does that simplify to ln(x/2)/2ln(sin(x/2))

[long marten]

Yes

You can replace everything in the limit equation

Then use expansion series

You know what expansion series is?

[cursive edge]

I havent learnt any of those lol

[long marten]

Maybe you can do it without expansion series

[cursive edge]

Im only at a gcse level so im not familiar with any of this

[long marten]

Just need to replace the log_blabla

[long marten]

I don’t know what gcse is because I’m French

[cursive edge]

What do i do from here

[cursive edge]

Im in year 11 of UK school so maybe it's smth similar there

[long marten]

Okay you did good and now you should use expansion series.. but you have to learn it first. It’s basically a formula that approximates a function with a polynomial at a certain point. For example here it’s point 0 since we want lim when x->0

[long marten]

Oh that’s nice!

[cursive edge]

So if I learn that I will be able to solve it?

[long marten]

But you can see many things right now. Lim ln(x) when x->0 is -infinity

[long marten]

Yes normally

Maybe try think like you usually do for basic limits

What is ln(x) when x is close to 0? -infinity

What is sin(x) when x is close to 0?

It’s 0

[cursive edge]

Ah that makes sense

And cos(x) when x is close to 0 is 1 then?

[long marten]

That’s the best you can do without knowing expansion series

[long marten]

Yes

[cursive edge]

Okay thank you very much 🙂

[long marten]

YOURE welcome!

[cursive edge]

.close

[mental ruin]

Solved

Post marked as solved by @cursive edge.

Use .unsolved if this was a mistake.

[cursive edge]

Wait actually how would I do anything with -infinity

Yk what its fine ima figure it out

[long marten]

That’s the point. Some functions in your problem will be closer to -infinity than the others

You must know the expansion series for sin(u) ln(u)

This will resolve the problem

[cursive edge]

Okay thx alot once again

[long marten]

The series expansion formula is in red

You must use that

You will replace all the complicated functions by polynomials which will be much much easier to determine the limit

[cursive edge]

Does the f'(x_0) mean the differentiated version of the function or the inverse

[long marten]

The differentiated version

[cursive edge]

Okay yeah I can do that

[long marten]

And x_0 is the point you want x close to for the limit

Your example is x->0

So x_0=0

[cursive edge]

What does the f^(n)(x_0) mean then

[long marten]

the nth derivative of your function at point x_0

[cursive edge]

Nth derivative?

What would I do if n = 2 then

Or does that not matter

[long marten]

Then you differentiate 2 times

[cursive edge]

Ah okay

[long marten]

It means f’’(x_0)

[cursive edge]

Ah okay

[long marten]

Nth just means if you want to differentiate n times

It is a generalization

[cursive edge]

So if f(x) = x³ + x² + x + 1 then f''(x) would be 6x + 2?

[long marten]

Yes

[echo gorge]

Sorry but are there other ways to solve it beside using Taylor ?

[long marten]

I don’t think so… i mean look at the monster

So many functions

[cursive edge]

How does the Taylor expansion actually let me reach a result if it goes on for infinity

[echo gorge]

Hmm.. maybe using loga(b)=lna/lnb ?

[long marten]

You inverted it should be loga(b)=ln(b)/ln(a)

[long marten]

You don’t have to do it until infinity

[cursive edge]

How do I know when to stop

[long marten]

Intuition. I would say you should stop at n=1

[echo gorge]

Ahh, yes yes my mistake

[cursive edge]

Okay

[echo gorge]

But i think maybe it will be easier

[long marten]

And yes we used that formula already

[cursive edge]

Do I have to include n = 0

[long marten]

Yes

It’s a sum so yes

[cursive edge]

Okay

[long marten]

f(x)=f(x_0)+f’(x_0)*(x-x_0)

If you stop at n=1

[long marten]

@echo gorge

[echo gorge]

I have a question that how do you know when we use Taylor?

I mean that's a real monster to me actually :)))

[long marten]

You use Taylor when you want to approximate a function with a polynomial at a certain « neighborhood »

[cursive edge]

And x_0 is just 0 so would it be f(x) = f(0)+f'(0)*(x-0)

[long marten]

For example close to 0, sin(x)=x

[long marten]

Yes

[echo gorge]

I didn't use Taylor and the result is 4, i don't know it's right or not

[echo gorge]

And L'hopital

[cursive edge]

What do i do with that then

[long marten]

Oh l’hôpital rule, well idk about this one. But the expansion series is more than enough for limits especially

[cursive edge]

f(0) is just 0 i assume

[long marten]

You replace every functions by their « taylor » forms you’ve found

In your limit

[cursive edge]

Ah

So i dont actually do anything with the Taylor yet

[echo gorge]

It's easier for me to use than Taylor 🥲

[long marten]

You must use the expansion series for every functions you have in your problem

For ln(x); 2ln(sin(x)); cos(x) etc..

[long marten]

I know…

[echo gorge]

Ahh, sorry for interrupting you guys, maybe I'm out now 🥲

[long marten]

But really it’s simple because it’s just a formula

[cursive edge]

Oh dont worry im learning alot from ur questions

[echo gorge]

Let me know if you find out the result then 🫶

[cursive edge]

Will do

[long marten]

YOURE not interrupting don’t worry ^^

[cursive edge]

Okay i will go eat then try that out afterwards

[long marten]

I will try to find the solution on my side

[cursive edge]

Okay 👍

[long marten]

Alright bon appétit!

[cursive edge]

Merci

[cursive edge]

Okay i am back

Wait i think I found out a method

[cursive edge]

Is the answer 1

.unsolved

[mental ruin]

Unsolved

Post marked as unsolved by @cursive edge.

Use .solved to mark as solved.

[echo gorge]

And i had 4 😭

[cursive edge]

I dont know wtf i did so mine is 100% wrong

[echo gorge]

I can't write for u 'cuz i turned off the light 🥲

It's 2.24 in VN 🥲

[cursive edge]

Wait I managed to find a a method and I got 4 aswell

I think thats good news

[long marten]

Hii

[long marten]

Yes it is 1

How did you do it?

[cursive edge]

Is it

[long marten]

Yes

[cursive edge]

Well i knew ln(0) approaches - infinity

[long marten]

Yeah?

[cursive edge]

So then I got ((-infinity/2(-infinity))/((-infinity/2(-infinity)))

And i canceled out the minus infinities

And got 1/2 ÷ 1/2

Which is 1

But I thought that was wronf

[long marten]

Well that’s a coincidence because normally that approach doesn’t work

[cursive edge]

Oh wow

[long marten]

For that you would have used the series expansion formula

I guess you’ve been lucky

[cursive edge]

Lol

The thing is

I asked chatgpt and it said 4

[long marten]

ChatGPT said 4? It can’t be..

[cursive edge]

Let me show u the working it gave me

[long marten]

Did it use the series expansions formulas?

[cursive edge]

It used some thing with (1+o(1))

.close

[mental ruin]

Solved

Post marked as solved by @cursive edge.

Use .unsolved if this was a mistake.

[cursive edge]

So its either 1 or 4 then

[long marten]

It is 1

Chat gpt is crazy

Just try it on a calculator if you want

[long marten]

Take x= 0,0000000001 for example

Put that on the calculator

[cursive edge]

@echo gorge also got 4

[long marten]

And you’ll see it will make 1

[cursive edge]

Okay lemme try

[long marten]

👍

[cursive edge]

I cant find sin^2 on my calc

[long marten]

No need

[long marten]

Look at the simplified formula of earlier

[cursive edge]

Okay

Oh

Now that I look at it

[long marten]

?

[cursive edge]

That method does give 1

[long marten]

Yup

[cursive edge]

Cuz x approaches 0

[long marten]

Told ya

[cursive edge]

Well thats odd

[long marten]

Chat gpt can sometimes be wrong

[cursive edge]

I also found an answer in the insta comment section and it gave 4

[long marten]

4??

Wait let me check if i converted the logs in the right base. Normally yes.

[long marten]

Well some steps have been dodged ?

Explain to me how they went from step 1 to step 2

There should be 2ln(sin(x)) somewhere

[cursive edge]

I have no clue

[long marten]

Ahhhh

Omg

I understand

I thought it was log_sin^2(x)*cos(x)

But no

Cos(x) was supposed to be INSIDE log

[cursive edge]

Ahhh

[long marten]

I misread es

Misreaded

I thought it was 2 functions lol

[cursive edge]

So it isnt 1 then

[long marten]

I over complicated things

No it isn’t

They should have put the parenthesis

[cursive edge]

Oh well it happens to the best of us lol

[cursive edge]

Thats true

[long marten]

Because i didn’t see

Well i could give you my steps for the new function which the limit is easier to calculate lol

[cursive edge]

Okay yes please

[long marten]

You basically use series expansion again

[cursive edge]

I want to see the series expansion method in action

[long marten]

Alright give me some time

[cursive edge]

Okay 👍

[long marten]

Sorry i had to go somewhere… but anyways here’s my steps. DL is an abbreviation of (« series expansion of »)

And i got the results by using series expansion. Anyways the problem is solved yayyy

[minor mantle]

!nosol

dont send the qn's answer

just guide the OP n let him do it by himself

[long marten]

The problem was already solved.
And we mutually shared our thinkings about the problem for many hours.
After that the OP knew the answer, i simply shared the steps that would lead to the solution because I and they wanted to.