Limits and Infinite Limits

[Ming1015]

Hello, I want to confirm for the definition of infinite limit. Does the presence of infinite limit mean that the limit of a certain function approaches (or can indirectly say 'equal to') infinity? Then, I also would like to ask if is it possible for the limit (finite) and infinite limit to exist together in a specific equation (or function).

 

[Dr.Peterson]

Hello, I want to confirm for the definition of infinite limit. Does the presence of infinite limit mean that the limit of a certain function approaches (or can indirectly say 'equal to') infinity? Then, I also would like to ask if is it possible for the limit (finite) and infinite limit to exist together in a specific equation (or function).

What definition were you given? Does it agree, for example, with this? https://tutorial.math.lamar.edu/classes/calci/infinitelimits.aspx

Properly, we don't say that a limit "approaches"; the limit "is" some value. That includes infinity, though you have to be careful since that isn't a number.

Are you asking about a finite limit and an infinite limit existing at the same place (how could that be?), or at different places (why could it not?). Perhaps you can give an example of what you have in mind.

 

[Ming1015]

What definition were you given? Does it agree, for example, with this? https://tutorial.math.lamar.edu/classes/calci/infinitelimits.aspx

Properly, we don't say that a limit "approaches"; the limit "is" some value. That includes infinity, though you have to be careful since that isn't a number.

Are you asking about a finite limit and an infinite limit existing at the same place (how could that be?), or at different places (why could it not?). Perhaps you can give an example of what you have in mind.

Yea Dr. Peterson, it's same as what it's described in the link you attached. Okay I see, so usually we will call it as 'limit is some value' instead of 'limit approaches some value'. Okay Dr Peterson, if the example is lim x->infinity [ (2x³+5)/(4x-3x²) ], if I'm asked to find whether the limit or infinite limit exists, and also find its value. What I know is the answer is negative infinity, so I have to say that the limit doesn't exist but the infinite limit does exist and its value is negative infinity, right? Then for the another example where the limit is finite, lim x->1 (x²-4x+3)/(x-1). The answer should be - 2, so I could say that the limit exists but the infinite limit doesn't exist, and its value is - 2. Is the way I write my answer correct?

 

[Dr.Peterson]

Yea Dr. Peterson, it's same as what it's described in the link you attached. Okay I see, so usually we will call it as 'limit is some value' instead of 'limit approaches some value'. Okay Dr Peterson, if the example is lim x->infinity [ (2x³+5)/(4x-3x²) ], if I'm asked to find whether the limit or infinite limit exists, and also find its value. What I know is the answer is negative infinity, so I have to say that the limit doesn't exist but the infinite limit does exist and its value is negative infinity, right? Then for the another example where the limit is finite, lim x->1 (x²-4x+3)/(x-1). The answer should be - 2, so I could say that the limit exists but the infinite limit doesn't exist, and its value is - 2. Is the way I write my answer correct?

Again, it may depend on details of the wording you are taught; I am not familiar with your wording, "whether the limit or infinite limit exists". I would just say "the limit is negative infinity", not " the limit doesn't exist but the infinite limit does exist and its value is negative infinity".

The page I referred to just says "evaluate the limit". Can you show us an example from your textbook that asks and answers this sort of question?

 

[Ming1015]

Again, it may depend on details of the wording you are taught; I am not familiar with your wording, "whether the limit or infinite limit exists". I would just say "the limit is negative infinity", not " the limit doesn't exist but the infinite limit does exist and its value is negative infinity".

The page I referred to just says "evaluate the limit". Can you show us an example from your textbook that asks and answers this sort of question?

Okay I see. I thought I should answer for the existence of limit or infinite limit because in the question I was asked to 'find the limit or infinite limit, if it exists'.

Anyway Dr. Peterson, these are some examples from my textbook with the answers and explanations beside them.

 

[Dr.Peterson]

Okay I see. I thought I should answer for the existence of limit or infinite limit because in the question I was asked to 'find the limit or infinite limit, if it exists'.

Anyway Dr. Peterson, these are some examples from my textbook with the answers and explanations beside them.

Good; their answers are what I would expect: There is no need to say "this is an infinite limit" when you've written [MATH]\infty[/MATH] as the answer, and it is wrong to say "the limit doesn't exist" when you've said the limit equals something.

In my mind (others may disagree) it is also unnecessary for them to distinguish "limit" from "infinite limit" in the question, which you have no control over! But I do notice that they say "if it exists", not "if either exists", so I don't think they are trying to make too big a distinction.

 

[Ming1015]

Good; their answers are what I would expect: There is no need to say "this is an infinite limit" when you've written [MATH]\infty[/MATH] as the answer, and it is wrong to say "the limit doesn't exist" when you've said the limit equals something.

In my mind (others may disagree) it is also unnecessary for them to distinguish "limit" from "infinite limit" in the question, which you have no control over! But I do notice that they say "if it exists", not "if either exists", so I don't think they are trying to make too big a distinction.

Well, I think I fully understood it. Thank you for your explanation, Dr. Peterson. But just an extra question from me, is there any difference if the question asks 'if it exists' with 'if either exists'?

 

[Dr.Peterson]

But just an extra question from me, is there any difference if the question asks 'if it exists' with 'if either exists'?

The point I was making is that the question implies they are asking only about one thing, "the limit". They are not asking about two different things, "the limit" and "the infinite limit".

 

[Ming1015]

The point I was making is that the question implies they are asking only about one thing, "the limit". They are not asking about two different things, "the limit" and "the infinite limit".

Okay, no problem. Thanks a lot for your assist, Dr. Peterson.

 

[Ming1015]

The point I was making is that the question implies they are asking only about one thing, "the limit". They are not asking about two different things, "the limit" and "the infinite limit".

Dr. Peterson, I have one more question regarding the existence of limits. For example, I take the limit x approaches 2 for function f(x) . Is it necessary that we should always check the values of lim x→2⁺ f(x) and lim x→2 ^- f(x) are the same first to prove that the limit exists?

 

[Dr.Peterson]

Dr. Peterson, I have one more question regarding the existence of limits. For example, I take the limit x approaches 2 for function f(x) . Is it necessary that we should always check the values of lim x→2⁺ f(x) and lim x→2 ^- f(x) are the same first to prove that the limit exists?

That depends on the nature of the function and how you are finding the limit. If the function has a piecewise definition, then you have to check explicitly for each direction; otherwise, whatever method you use commonly takes that into account implicitly (e.g. simplifying to a continuous function by "filling in a hole").

 

[Ming1015]

That depends on the nature of the function and how you are finding the limit. If the function has a piecewise definition, then you have to check explicitly for each direction; otherwise, whatever method you use commonly takes that into account implicitly (e.g. simplifying to a continuous function by "filling in a hole").

Okay noted, I got it already. Thanks ya?.