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Numerical

Member

Registered: 2016-02-08

Posts: 2

Why Rationalize Infinite Limits

Hello MathisFun forummers,

I stumbles accross this excercise:

and I was thinking why should I rationalize it? The book is very summier about explaining why it has to be rationalized and says it is also a possibility. In the answers it gets rationalized and I understand that I have to do it too, but I don't see the why!?

Thanks in advance

Last edited by Numerical (2016-02-08 06:21:08)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

The best response is can you do it another way and get the right answer?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Why Rationalize Infinite Limits

Aren't there several ways to solve it? I think the question might be why does it need to be solved. I would say it is just a formality to clarify what the function is getting closer to in that direction. Also, I have heard infinite limits and sums show up all the time in quantum physics.

Numerical

Member

Registered: 2016-02-08

Posts: 2

Re: Why Rationalize Infinite Limits

The best response is can you do it another way and get the right answer?

Well my goal with inifinite limits - actually all limits - is evaluating what happens when you get closer and closer to a certain x and by rationalizing it I get a more basic function which shows me what truly happens at that point.

Maybe the better question is: If I rationalize this function the domain and range of a normal function is lost and a new function appears how can it be that it still hold the proper value that?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

Maybe you had better show me exactly what you do with that limit and we can go from there.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Why Rationalize Infinite Limits

Hello MathisFun forummers,

I stumbles accross this excercise:

and I was thinking why should I rationalize it?

You are NOT rationalizing it; you are doing the reverse!

To rationalize is to eliminate the square-root signs in the denominator. Here you want to put the square-root signs back in the denominator!

And why do you that? Because it's the only way to solve limit problems of this sort!

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anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Why Rationalize Infinite Limits

Hello MathisFun forummers,

I stumbles accross this excercise:

and I was thinking why should I rationalize it?

You are NOT rationalizing it; you are doing the reverse!

To rationalize is to eliminate the square-root signs in the denominator. Here you want to put the square-root signs back in the denominator!

And why do you that? Because it's the only way to solve limit problems of this sort!

Not exactly true...

Maybe a bit more complicated than the anti-rationalization method, but much more generalizable.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

A computational attack would Taylorize the expression:

where the answer stands out clearly.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Why Rationalize Infinite Limits

... Which is basically the same thing I did above.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

Are you sure?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Why Rationalize Infinite Limits

Relevant line:

Here, I applied Taylor (or more specifically Maclaurin), I just didn't state it explicitly.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

Look at the RHS of post #8

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Why Rationalize Infinite Limits

Taylor does not exactly make sense at infinities...

And, before we go any further, you'll have to agree that that would lead to the incorrect answer of 2.

Last edited by anonimnystefy (2016-02-12 19:07:23)

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

Exact a Mundo! Now what went wrong?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Why Rationalize Infinite Limits

I am not sure what you even tried to do.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

Tried to expand that at negative infinity.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Why Rationalize Infinite Limits

I notice Mathematica can't actually expand around negative infinity.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Why Rationalize Infinite Limits

That is correct!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.