Math Is Fun Forum

ohlookabirdie

Member

Registered: 2011-08-23

Posts: 3

i have 3 Analysis questions...

i have 3 Analysis questions whxch look simple to me, but for the life of me i cannot work out... i think i must just be being stupid, or having an off day :S, anyway they are as follows:

1) Use the triangle inequality to prove that for all a, b ∈ R,

|a-b| ≥ |a| - |b|.

2) Find the limit of Lim(√(n² + n) -n) from n to ∞.

(here i know the answer to be 1/2 but i dont know how to get it)

3) Use standard results to show that the function

f(x) = {x² for x ∈ (-∞,1), 1 for x ≥ 1}

is continuous on R.

(R is the real numbers, i dont know how to do the special R for it)

sorry if my formating is off, im not good with typing math

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: i have 3 Analysis questions...

Hi ohlookabirdie;

For #2

I would do it like this:

Pull out an n^2 from the radical.

Remember the sqrt root is a principal value and is always positive.

We can eliminate the absolute value because our limit is approaching positive infinity.

Now use the conjugate:

This multiplication only looks tough and you can use a trick to simplify it.

The radical in the denominator is easy now,

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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.

ohlookabirdie

Member

Registered: 2011-08-23

Posts: 3

Re: i have 3 Analysis questions...

i had almost go there thanks

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I may be a nerd, but i can still be artistic, so here is a link to my photo gallery

Tom - "it looks like you've been keepin yourself busy with an attempt to photo every bit of wildlife in the country"

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: i have 3 Analysis questions...

Your welcome but I only did 1 out of 3.

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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.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: i have 3 Analysis questions...

#1).  Why is this a triangle with possible negative values?  Are they vectors?  Can you state the triangle inequality theorem you need to use?
wiki is unclear on this, though the answer does seem obvious, as you can see, but I know you need a rigourous proof.

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igloo myrtilles fourmis

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: i have 3 Analysis questions...

hi JEF

i don't think there is an actual triangle here.i think he's referring  to the triangle inequality for two vectors:

this is one proof:
write x=x-y+y
so x= (x-y)+y

using the triangle inequality we get:

from there we get:

which is what we wanted to prove.

hope this helps

Last edited by anonimnystefy (2012-01-25 22:01:40)

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