Epsilon in calculate limits...

[shahar]

Why we need in every limit arithmetics to note that epsilon greater than zero?

 

[lev888]

Why we need in every limit arithmetics to note that epsilon greater than zero?

Please post an example.

 

[shahar]

Proof that a_n ---> L and b_n ---> K when Ca_n * Cb_n --> CLK.
C = constant.

 

[lev888]

Proof that a_n ---> L and b_n ---> K when Ca_n * Cb_n --> CLK.
C = constant.

Don't see where the epsilon is mentioned. Please post complete text you are asking about.

 

[shahar]

A and b are series that converge to limit. a and b have a limits K and L, appropriately. Show that by multiplying the limits by C the will give the covergation of the limit C²KL

 

[lev888]

A and b are series that converge to limit. a and b have a limits K and L, appropriately. Show that by multiplying the limits by C the will give the covergation of the limit CKL

Don't see where the epsilon is mentioned. Please post complete text you are asking about.

 

[pka]

Why we need in every limit arithmetics to note that epsilon greater than zero?

In the definition of limits, the epsilon is a measure of distance, a measure of closeness. Distances are all non-negative.

 

[shahar]

In the definition of limits, the epsilon is a measure of distance, a measure of closeness. Distances are all non-negative.

Why I will get a grade that has lost points if I not mention the expression: epsilon is greater than zero?

 

[pka]

Why I will get a grade that has lost points if I not mention the expression: epsilon is greater than zero?

Unless \(\varepsilon>0\) it cannot be used as a measure of closeness.
To say \(|x-x_0|<\varepsilon\) is shorthand for \(\{x|x_0-\varepsilon<x<x_0+\varepsilon\}\).
That is, \(x's\) within an \(\varepsilon\) distance of \(x_0\). That does not work unless \(\varepsilon>0\).

 

[nasi112]

Unless \(\varepsilon>0\) it cannot be used as a measure of closeness.
To say \(|x-x_0|<\varepsilon\) is shorthand for \(\{x|x_0-\varepsilon<x<x_0+\varepsilon\}\).
That is, \(x's\) within an \(\varepsilon\) distance of \(x_0\). That does not work unless \(\varepsilon>0\).

I have been for years struggling and avoiding proving things with epsilon. Later, found it is a simple thing. Most students now have the same problem.

shahar,
your topic in this thread is very important

 

[Steven G]

Distance is never negative and we certainly do not want ε=0.