Show that: lim (x+y)=o as x and y approach zero; using the epsilon-delta definition
Also, show that: lim f(x)=1 as x approaches zero; using the epsilon-delta definition.
**Note: x is a vector in this case, with a right arrow going over it. It is not just "x".
We have a function f(x,y) = (x+y), and we need to show that:
Lim f(x,y) = Lim (x + y) = 0
x -> 0 x -> 0
y -> 0 y -> 0
First we need to define and understand what is meant by limit.
A function f(x,y) tends to limit L as x tends to a, and y tends to b if and only if:
For each real number epsilon > 0, there can be found a real number delta > 0 such that the following statement is always true:
|x - a| < delta and |y - b| < delta and (x,y) is not equal to (a,b) (1)
| f(x,y) - L | < epsilon (2)
So for our problem, we can write:
Lim (x + y) = 0
x -> 0
y -> 0
if and only if for each epsilon > 0, we can find a delta > 0 such that the following statement is true:
If: |x| < delta and |y| < delta and (x,y) is not equal to (0,0) (3)
then: |x + y - 0| < epsilon (4)
So the challenge we have to meet is this: If someone was to give us any real number epsilon > 0 (no matter how small) we must be able to assure them that we will definitely be able to give them back a delta > 0 such ...
This shows how to find limits using epsilon-delta.