Understanding Limits: Real Numbers or Infinity?

  • Thread starter buckr02
  • Start date
  • Tags
    Mean
In summary, the conversation discusses the concept of limits and whether the limit as x approaches a real number is infinity or undefined. It is mentioned that in the extended reals, infinity is included and is commonly used as shorthand for the limit being undefined.
  • #1
buckr02
1
0
I can't believe I'm asking this, because I should know this answer but I'm now doubting myself.

If a function goes to infinity as x approaches some real number, would we say the limit as x approaches that number is infinity or would we say that it does not exist?

Doesn't the limit always have to be a real number, but infinity isn't one?

So for example:
the limit as x approaches 0 from the right of 1/x is infinity or not defined?
 
Physics news on Phys.org
  • #2
It is a question of whether you are in the reals or the extended reals. The latter includes infinity.
 
  • #3
Generically people will say the limit equals infinity. Indeed infinity is not a real number, but it is useful to use the shorthand of saying the limit equals infinity, as this is more specific than just saying that the limit is undefined.
 

FAQ: Understanding Limits: Real Numbers or Infinity?

What is the concept of lim=infinity?

The concept of lim=infinity means that the limit of a function or sequence approaches infinity as the input or index value approaches a certain value or goes to infinity. In other words, the function or sequence grows without bound as the input or index value increases.

How is lim=infinity different from a regular limit?

The main difference between lim=infinity and a regular limit is that lim=infinity indicates that the limit is unbounded, while a regular limit can approach a specific finite value or may not exist at all.

How is lim=infinity used in mathematics?

Lim=infinity is used to describe and analyze the behavior of functions or sequences as the input or index value approaches infinity. It is an important concept in calculus and is used to evaluate limits, derivatives, and integrals of functions.

What are the properties of lim=infinity?

Some properties of lim=infinity include linearity, which means that the limit of a sum or difference of two functions is equal to the sum or difference of their limits. It also follows the product rule, quotient rule, and power rule for limits. Additionally, it has the property of squeeze theorem, which states that if two functions have the same limit at a point, then a third function squeezed between them also has the same limit at that point.

Can lim=infinity have a negative value?

No, lim=infinity cannot have a negative value. The concept of lim=infinity only indicates that the limit is unbounded and approaches infinity, but it cannot be a negative infinity. However, a function or sequence can have a limit of negative infinity, which indicates that it approaches negative infinity as the input or index value approaches a certain value or goes to infinity.

Similar threads

I A question about limits and infinity
Replies
7
Views
2K
B Difficulty with understanding whether 1/n converges or diverges
Replies
11
Views
3K
I To What Extent Can You Manipulate Limits?
Replies
9
Views
495
What is the limit of x^(1/log(x)) as x approaches infinity?
Replies
10
Views
5K
MHB Find the limit as x goes to infinity
Replies
5
Views
2K
B How can hyperreal numbers make infinitesimals logically sound in calculus?
Replies
24
Views
4K
I Determining the Rate at Which Functions approach Infinity
Replies
3
Views
8K
I Precise intuition about limits and infinitesimals
Replies
4
Views
2K
I Can indeterminate limits always be evaluated?
Replies
10
Views
1K
B Dividing by infinity, exactly, finally!
2
Replies
40
Views
4K

Hot Threads

Recent Insights

Back
Top