- #1

MelanieSwan

- 11

- 0

## Homework Statement

true or false. prove or give counter example:

1. with x approaches to infinity suppose that lim f(x) = L and lim g(x) = M and f(x)< g(x), then L< M.

2. with x approaches to infinity suppose that lim f(x) = L and lim g(x) = M and f(x)<= g(x), then L<=M

for all x in D

## Homework Equations

## The Attempt at a Solution

1. I know that this is wrong and one counter example I can think of is: f(x) = 2/|x| and g(x) = 3/|x| so for all non zero x, f(x) < g(x) but their limits are both zero.

2. I came up with this. I wonder if it's correct?

for every e > 0 and e is very very small then there exists m1 in N such that for every x>= m1 we have: |f(x) - L| <e rewrite as e> f(x) - L> -e (1)

similarly there exists m2 in N such that for every x>=m2 we have |g(x) - M| <e rewrite as -e< g(x)-M <e (2)

subtract (1) from (2) we have:

-2e < g(x) - f(x) + L- M) < 2e.

we only care about the right part of the inequation above so:

L-M < 2e +f(x) -g(x)

since f(x) - g(x) <= 0, there always exists e positive but very very small so that 2e + f(x) - g(x) <= 0

so L-M <= 0

so L <= M as desired.