Understanding Limits: Defining and Applying the Formal Definition

In summary, the conversation discusses the formal definition of a limit and how it can be misunderstood. The main error in the application of the definition is that someone else chooses the value of epsilon, and from that, a corresponding value of delta needs to be found. Using an example, it is shown that this cannot always be done and therefore, the claim that any value of L is the limit for any function at an arbitrary x0 is incorrect.
  • #1
alexjean
2
0
Homework Statement

I know what a limit is and I understand the idea behind it, but I am misunderstanding something in the formal definition of a limit.

DEFINITION
Let f(x) be defined on an open interval about x0, except possibly at x itself. We say that the limit of f(x) as x approaches x0 is the number L and write

limx → x0 f(x) = L

if, for every number ε > 0 there exists a corresponding number δ > 0 such that for all x,
0 < abs(x – x0) < δ
=>
abs(f(x) – L) < ε



The attempt at a solution

For example, take f(x) = x2

I could, incorrectly, assert that limx → 2 f(x2) = 1

Now, δ > abs(x - 2) > 0
and ε > abs(f(x) - 1)

If:
1) this holds true for all values of x (except x = x0 = 2), and,
2) for every possible value of epsilon some value of x which satisfies the above statement for both ε and δ exists,
Then:
1 is the limit of x2 as x approaches 2.


As an example, I pick x = 3. So,
δ must be > 3
and
ε must be > 8

I can continue and choose arbitrary values of x, none of which seem to be a problem.

Likewise, for any value of epsilon I can think of, I can find an value of x which satisfies the inequality ε > abs(f(x) - 1) which also satisfies δ > abs(x - 2) > 0.
ex: when ε > .01 I could have x = .999. This x also satisfies δ > abs(x - 2) > 0
(as far I can tell, it's impossible not to satisfy gamma, as long as x isn't equal to x0)

By that reasoning, the claim I made above is true and limx → 2 f(x2) = 1



By the reasoning I'm using with the formal definition, I can state that any value of L is the limit for any function at an arbitrary x0 (given that the function has values infinity close to L at some point in time). This is clearly incorrect. Would anyone please point out the error in my application of the definition of a limit?

Thank you.
 
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  • #2
alexjean said:
Homework Statement

I know what a limit is and I understand the idea behind it, but I am misunderstanding something in the formal definition of a limit.

DEFINITION
Let f(x) be defined on an open interval about x0, except possibly at x itself. We say that the limit of f(x) as x approaches x0 is the number L and write

limx → x0 f(x) = L

if, for every number ε > 0 there exists a corresponding number δ > 0 such that for all x,
0 < abs(x – x0) < δ
=>
abs(f(x) – L) < ε
The attempt at a solution

For example, take f(x) = x2

I could, incorrectly, assert that limx → 2 f(x2) = 1

Now, δ > abs(x - 2) > 0
and ε > abs(f(x) - 1)

If:
1) this holds true for all values of x (except x = x0 = 2), and,
2) for every possible value of epsilon some value of x which satisfies the above statement for both ε and δ exists,
Then:
1 is the limit of x2 as x approaches 2.As an example, I pick x = 3. So,
δ must be > 3
and
ε must be > 8

I can continue and choose arbitrary values of x, none of which seem to be a problem.

Likewise, for any value of epsilon I can think of, I can find an value of x which satisfies the inequality ε > abs(f(x) - 1) which also satisfies δ > abs(x - 2) > 0.
ex: when ε > .01 I could have x = .999. This x also satisfies δ > abs(x - 2) > 0
(as far I can tell, it's impossible not to satisfy gamma, as long as x isn't equal to x0)

By that reasoning, the claim I made above is true and limx → 2 f(x2) = 1
By the reasoning I'm using with the formal definition, I can state that any value of L is the limit for any function at an arbitrary x0 (given that the function has values infinity close to L at some point in time). This is clearly incorrect. Would anyone please point out the error in my application of the definition of a limit?

Thank you.

Your error is that someone else chooses ##\epsilon##, and from that you have to find a δ > 0 so that, when |x - 2| < δ, |f(x) - L| < ##\epsilon##.

Using your example, where you postulate that ##\lim_{x \to 2}x^2 = 1##, I will say that ##\epsilon = .01##. You now have to find a pos. number δ such that for each x ##\in## (2 - δ, 2 + δ), then x2 ##\in## (.99, 1.01).

Sketch a graph of the function y = x2, and you'll see that this can't happen.
 

What is the definition of a limit?

The definition of a limit is a fundamental concept in calculus that describes the behavior of a function as the input approaches a certain value. It is typically denoted by the symbol lim and is used to determine the value that a function approaches or "tends to" as the input gets closer and closer to a specific value.

How is a limit formally defined?

The formal definition of a limit is as follows: Given a function f(x) and a value c, the limit of f(x) as x approaches c is equal to L if for any positive number ε, there exists a positive number δ such that if x is within δ units of c (but not equal to c), then f(x) is within ε units of L. This is represented mathematically as lim x→c f(x) = L.

Why is the concept of a limit important?

The concept of a limit is important because it allows us to understand the behavior of a function at a specific point, even when the function is not defined at that point. It also allows us to find the slope of a tangent line to a curve, which is essential in many real-world applications such as physics, engineering, and economics. Additionally, limits are a crucial component in the development of more advanced mathematical concepts, such as derivatives and integrals.

What are the two types of limits?

The two types of limits are one-sided limits and two-sided limits. A one-sided limit, also known as a directional limit, is when the input approaches the specified value from only one side. A two-sided limit, on the other hand, is when the input approaches the specified value from both the left and right sides.

How is a limit evaluated?

A limit is evaluated using various techniques, such as direct substitution, factoring, rationalizing, and using special limit laws. Additionally, L'Hopital's rule can be used to evaluate certain types of limits involving fractions, exponentials, and logarithms. In some cases, a limit may be evaluated graphically by looking at the behavior of the function near the specified value. In more complex cases, numerical methods such as using a graphing calculator or computer software may be necessary to approximate the limit.

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