Understanding Limit Definition and the Role of Inequalities in Calculus

In summary, the alternative definition of a limit uses inequalities that are stricter than the standard definition. This difference makes the limit hold for undefined functions.
  • #1
Yoni V
44
0

Homework Statement


It is not exactly a homework question, but why does the definition of a limit use strict inequalities as follows:
if 0 < |x - a| < δ, then |f(x) - l| < ε
rather than weak inequalities, for example
if 0 < |x - a| < δ, then |f(x) - l| ≤ ε

Could the addition of the equality option make a difference?

Homework Equations

The Attempt at a Solution


I tried thinking of functions that would yield different limits to the limit produced by the formal definition, but couldn't find any.
I also tried to rule it out somehow with formal deduction, but couldn't.
Any hints or ideas?

Thanks
 
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  • #2
Ok, I realized that the fact I couldn't disprove it is because it indeed holds.
It might not be as nice to be the definition, but given the definition,
if 0 < |x - a| < δ, then |f(x) - l| ≤ ε
implies that the limit of f is l.
 
  • #3
We proved early in our analysis course as a "recreational" activity that the two are, in fact, equivalent statements, but we just agreed to use the strict inequality.
 
  • #4
It's obvious that if ##\lim_{x\to a}f(x)=l## in the sense of the standard definition, then ##\lim_{x\to a}f(x)=l## in the sense of the alternative definition.

Suppose that ##\lim_{x\to a}f(x)=l## in the sense of the alternative definition. Let ##\varepsilon>0##. Let ##\delta>0## be such that the following implication holds for all ##x\in\mathbb R##,
$$0<|x-a|<\delta~\Rightarrow~|f(x)-l| <\frac\varepsilon 2.$$ (Our assumption ensures that such a ##\delta## exists). For all ##x\in\mathbb R## such that ##0<|x-a|<\delta##, we have ##|f(x)-l|<\frac\varepsilon 2<\varepsilon##. This implies that ##\lim_{x\to a}f(x)=l## in the sense of the standard definition.
 
  • #5
Thanks for your responses.
Could you refer me to the analysis course you mentioned?

Edit: Oh, I guess you meant a course you took elsewhere, not some section here in PF. Nvm...
 

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 its input values approach a particular value. It is defined as the value that a function approaches as its input values get closer and closer to a specific value.

How is the limit of a function written mathematically?

The limit of a function is written mathematically as lim f(x) = L, where the limit of the function f(x) approaches the value of L as x approaches a specific value. This notation indicates that the limit is a single number, rather than a function.

What is the difference between a left-sided and right-sided limit?

A left-sided limit refers to the behavior of a function as its input values approach a specific value from the left side, while a right-sided limit refers to the behavior of a function as its input values approach a specific value from the right side. The only difference between the two is the direction from which the input values approach the specific value.

How do you determine if a limit exists?

A limit exists if the left-sided and right-sided limits of a function are equal at a specific value. In other words, if the function approaches the same value from both sides, the limit exists. However, if the left-sided and right-sided limits are not equal, the limit does not exist.

Can a function have a limit at a point but not be continuous?

Yes, a function can have a limit at a point but not be continuous. This can occur if there is a "hole" or a "jump" in the function at that point, which means that the function is not defined at that specific value. In order for a function to be continuous, it must have a limit at every point and the limit must be equal to the value of the function at that point.

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