Epsilon delta to N & M Definition

In summary, for a simple finite limit, we can make the y bounds of epsilon constrain the limit by making delta really small. To prove that the limit of sqrt(x) as x approaches infinity is infinity, we can set up the statement: for every positive N, there exists a positive M such that when x is greater than M, sqrt(x) is greater than N. This can be achieved by setting M equal to N squared.
  • #1
prasannaworld
21
0
Okay for a simple finite limit: e.g.
lim (3x) = 3
x->1

in the end I say:

"Therefore for every |x - 3| < delta, there exists an epsilon such that |3x-3| < epsilon"

Hence I can make delta really really small and the y bounds of epsilon will constrain the limit.



So let's come to the example I saw in an article
lim (SQRT(x)) = INF
x-INF

Okay so:
x > N - x is greater than any positive integer
Match N with M^2
x > M^2
SQRT(x) > M

Okay so how will I make my statement?
 
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  • #2
so you are trying to prove that:

[tex] \lim_{x\rightarrow \infty}\sqrt{x}=\infty[/tex] right?

What we want to prove is that [tex] \forall N>0, \exists M>0[/tex] such that whenever

[tex] x>M, \sqrt{x}>N[/tex]


By observation we have, as you pointed out:[tex]\sqrt{x}>N=> x>N^2[/tex]

so our statement would be

[tex] \forall N>0, \exists M=N^2>0[/tex] such that whenever

[tex]x>M=N^2=>\sqrt{x}>N[/tex]
 
  • #3
Yes. That is what I wanted. Still I think the best way for me to get this is convince myself by trying to prove a false limit (I obviously should not be able to...)
 

FAQ: Epsilon delta to N & M Definition

What is the Epsilon Delta to N & M Definition?

The Epsilon Delta to N & M Definition is a method used in calculus to formally define the limit of a function. It involves choosing a small value, epsilon, and finding a corresponding value, delta, for which the function will remain within epsilon units of the limit for all inputs within delta units of the limit point.

Why is the Epsilon Delta to N & M Definition important?

The Epsilon Delta to N & M Definition is important because it provides a rigorous and precise way to define the limit of a function. It allows us to understand the behavior of functions near a specific point, which is essential in many fields of mathematics and science.

How is the Epsilon Delta to N & M Definition used?

The Epsilon Delta to N & M Definition is used to prove the existence of a limit and to find its numerical value. By choosing appropriate values for epsilon and delta, we can determine the behavior of a function at a specific point and make predictions about its behavior at other points.

What are the main components of the Epsilon Delta to N & M Definition?

The main components of the Epsilon Delta to N & M Definition are the limit point, epsilon, and delta. The limit point is the point at which we are trying to find the limit of the function. Epsilon represents the desired margin of error or closeness to the limit, and delta represents the corresponding distance from the limit point within which the function must remain within epsilon units of the limit.

How is the Epsilon Delta to N & M Definition related to other methods of finding limits?

The Epsilon Delta to N & M Definition is a more formal and rigorous approach to defining limits compared to other methods such as the graphing, algebraic, or numerical methods. It allows for a more precise understanding of the behavior of a function near a specific point and can be used to prove the existence of a limit in cases where other methods may not work.

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