I think there was a proof using Rolles Theorem. Please tell me that one if you can find it on the forum or if you know it.

Please tell me the formal Proof of L'Hospital Rule. I remember some days ago it was posted by someone but I can't find it.

2nd one is really easy.

The critical points are the ones where sin x = 1 or sin x = -1. Here the limit is 1

for -1< sin x< 1 the limit is 0. So the points of discontinuity are:

Question 3)

An Alternate Method:

Question 3)

QED

Hey If D is zero then the quadratic is necessarily a perfect square (I am not sure I understand what you mean by a perfect square) but the inverse may not be true.

Example. If triangle ABC covers triangle PQR then it is not necssary that Triangle PQR covers triangle ABC.

You can try proving it using LMVT (lagrange man value theorem)

@hsbhatt sir

Please examine my explanation and tell me if I am wrong:

It is discontinuous at rational numbers as proved by you

Now we know between every 2 rational numbers there is an irrational number and irrational numbers in (0,1) will look like:

0.1625388946294738...........................till infinite number of decimals and non repeating

Now if an rational number approaches this irrational number then it will look like:

0.1625388946294738........................... but it will end at some point (unlike the irrational number above)

so if we express this in the form of p/q where p and q are integers, q != 0 and gcd of p and q is 1 then p------>infinty and q also -------> infinty

So 1/q tends to zero. So since   and function = 0 at irrational numbers it is continuous at irrational numbers

@shinee

Please check once more urs is coming 9 calls (same way that I used)

1 1 1 1 1 1     (denotes each woman knows one gossip)

2 2 1 1 1 1   

2 2 2 2 1 1      (note: woman number 1 and number 3 have 2 different gossip each)

2 2 2 2 2 2

2 2 4 2 4 2

2 6 6 2 4 2

And 4 more calls

Total 9 calls

I understood how you proved that it is discontinuous at rational numbers. Its a superb proof.

but I did not understand your other post

Any book is fine as long as it has a lots of questions (for practice)

For theory I don't know (I follow my sir's notes and goiit for doubts)

@shinee

Then your video cam does not have high enough frame rates.

If you have seen on discovery and national geographic they use high speed cameras (with fram rates > 1000 frames per second!) to observe bullets being shot from a gun.

@ naughtygal

coprime means that the GCD of p& q is 1. eg. 3 and 4 are coprime. 19 and 102 are coprime

@ ankurgupta91

how is it continuous at irrational.

@hsbhatt sir

"So, from this, if you consider a sequence of rational numbers that in the limit is an irrational number, the limit of the sequence is definitely non-zero"

Sir how can you say it will be non zero.

"The numerator  of the bracketed expression is an odd number and the denominator is 2ⁿ.



So, again, we see that the denominator does not tend to q for the members of the sequence."

Sir please could you explain this part I did not understand it.

@sachinguptaiit 

but q is not constant it mqaybe q is very large so 1/q ----------> 0

I'm not saying its right or wrong unless you post the explanation