Understanding Limits in Calculus: Exploring the Concept of Neighborhoods

In summary, the expression "the limit of f(x) is b when x->a" can be thought of as saying that for every x that is very close to a but not equal to a, there is an f(x) that gets closer to b. However, this is not precise enough for a mathematical definition. The idea of \lim_{x\to a}f(x)=b is that you can make f(x) as close as you want to b by choosing x close enough, but not equal to, a. This means that the distance |f(x)-b| can be made smaller than any given positive number. This is also known as the epsilon delta method.
  • #1
daniel_i_l
Gold Member
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Is it correct to think about the expresion:
"the limit of f(x) is b when x->a" as saying that for every x that's very close to a but not a (in the deleted neighborhood of a) there is a f(x) that's very close to b (in the neibourhood of b) - or is that not precise enough?
 
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  • #2
It's definitely not precise enough for a mathematical definition. What do you mean by 'very close to a'. What is 'very close'?
The idea of [itex]\lim_{x\to a}f(x)=b[/itex] is that you can make f(x) as close as you want to b by choosing x close enough (but not equal to) a. By close I mean that the distance |f(x)-b| can be made as small as we want. How small? Smaller than any given positive number.

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html
 
  • #3
Might want to use 'gets closer' instead of 'is close to'

Thats pretty much the epsilon delta method.
 
  • #4
Thanks guys for making that clear.
 

What is a limit expression?

A limit expression is a mathematical concept used to describe the behavior of a function as its input values approach a particular value. It is typically denoted by the symbol "lim" and is used to find the value of a function at a specific point or determine if the function has a limit at that point.

What is the purpose of using limit expressions?

Limit expressions are used to understand the behavior of a function at a specific point or as the input values approach a particular value. They are essential in calculus and are used to calculate derivatives, integrals, and other important mathematical concepts.

How do you evaluate a limit expression?

To evaluate a limit expression, you can either use algebraic methods or graphical methods. Algebraically, you can use various limit laws and rules to simplify the expression and determine the limit. Graphically, you can use a graphing calculator or plot the function to visually see the behavior as the input values approach the desired value.

What are some common types of limit expressions?

Some common types of limit expressions include finite limits, infinite limits, one-sided limits, and trigonometric limits. Each type has its own set of properties and rules for evaluation.

What are some tips for solving difficult limit expressions?

Some tips for solving difficult limit expressions include using algebraic manipulation to simplify the expression, using L'Hopital's rule, and plugging in different values for the variable to see if there is a pattern. It is also helpful to familiarize yourself with common limit properties and rules to make the evaluation process easier.

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