Math Is Fun Forum

Devonne Lee

Guest

Can Someone Explain the Following Equation

Compare the orders of growth of:

and

Solution:

Can someone explain to me how the solution works?

Particularly, I would like to know why

- Is it ok to just move \frac{1}{2} right beside the limit (ie. what does that mean?)
- How did the whole thing equal to \frac{1}{2} in the end

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Can Someone Explain the Following Equation

Hi Devonne Lee;

You can always move a constant to the outside of the limit.

How did the whole thing equal to

in the end

Because that limit evaluates to 1.

---

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.

Devonne Lee

Guest

Re: Can Someone Explain the Following Equation

Hey bobbym,

Sorry, I'm really dumb when it comes to these things so please bear with me. Can you elaborate on that point? Do you mean moving the constant

to the outside of the limit (which is

)? I'm not really sure what that means.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Can Someone Explain the Following Equation

Hi;

The 1 / 2 is a constant. It is not part of the limit process which is examining what happens when n approaches infinity. Well 1 / 2 does change no matter what n does. So it can be pulled out of the limit because it is not dependent on n.

---

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.

Devonne Lee

Guest

Re: Can Someone Explain the Following Equation

Oh, I saw your edit. I think I understand it now! Thanks!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Can Someone Explain the Following Equation

Did you see why the limit is 1? It can even be understood intuitively.

Welcome to the forum!

---

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.

Devonne Lee

Guest

Re: Can Someone Explain the Following Equation

Yes, I understand why the limit is 1. As n gets larger and larger, the result of the function (is the what that thing is called?) gets closer to 1. The part that confused me the most is moving the constant 1/2 to the side.

This pretty much cleared it up for me:

The 1 / 2 is a constant. It is not part of the limit process which is examining what happens when n approaches infinity

Do I take it then that everytime I compute limits, I should move all constants "to the side" and multiply it later right after getting the actual limit?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Can Someone Explain the Following Equation

Hi Devonne Lee;

Generally yes. But math is funny, sometimes it is helpful not to. Sometimes it is helpful not to simplify something. Sometimes to see the pattern you do not simplify. But again, yes you are always allowed to do that.

---

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.

Devonne Lee

Guest

Re: Can Someone Explain the Following Equation

Ah, I understand what you mean. Thank you very much!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Can Someone Explain the Following Equation

Your welcome and I hope the answer was understandable.

---

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.