Integrate
Junior Member
- Joined
- May 17, 2018
- Messages
- 107
Whoever made that clear to you did you an enormous disservice. It is not the case.It was made very clear to me that limits never equal the output value.
James Stewart 7th edition
I think you meant [imath]\mathop {\lim }\limits_{x \to {x_0}} f(x) = f\left({{x_0}}\right)[/imath]If a function [imath]f[/imath] is continuous at [imath]x_0[/imath] then [imath]\mathop {\lim }\limits_{x \to {x_0}} f(x) =\red{ \left( {{x_0}} \right)}[/imath]
Check in your book for the definition of continuity. There should also be a theorem that says a polynomial is continuous everywhere.View attachment 32416
My understanding is that limits do not equal the value of a function. They only show the behavior AROUND the value of a function. It was made very clear to me that limits never equal the output value.
Why is this not the case here?
James Stewart 7th edition
Why can they not substitute?Whoever made that clear to you did you an enormous disservice. It is not the case.
If a function [imath]f[/imath] is continuous at [imath]x_0[/imath] then [imath]\mathop {\lim }\limits_{x \to {x_0}} f(x) = \left( {{x_0}} \right)[/imath]
Now what you may have been told that in evaluating limits do not simply substitute.
i.e. [imath]\mathop {\lim }\limits_{x \to {a}} f(x)[/imath] may well exist but it may be that [imath]\mathop {\lim }\limits_{x \to {a}} f(x)\ne f(a)[/imath]
[imath][/imath][imath][/imath][imath][/imath]
Thank you and yes this is my book.Check in your book for the definition of continuity. There should also be a theorem that says a polynomial is continuous everywhere.
Together, these answer the question and correct your misunderstanding.
As to the definition of limits, here is a snippet from what may be your book:
This is probably what you misinterpreted. Do you see the difference?
Correct. But if you know the function is continuous, then you can just use f(a).Thank you and yes this is my book.
Let me see if I understand this then.
If a limit and a function equal each other [then] it is continuous.
BUT for a limit to exist it does not need to actually equal f(a). Just that both sides approach f(a).
Now what you may have been told that in evaluating limits do not simply substitute.
i.e. [imath]\mathop {\lim }\limits_{x \to {a}} f(x)[/imath] may well exist but it may be that [imath]\mathop {\lim }\limits_{x \to {a}} f(x)\ne f(a)[/imath]
You can substitute, under the right conditions, namely when you know the function is continuous.Why can they not substitute?
Lets take a really simple example.Why can they not substitute?
Excellent example. Thank you.Lets take a really simple example.
[imath]f(x)=|x-1|+3\text{ if }x\ne 1~\&~4\text{ if }x=1 [/imath]
In this example [imath]\mathop {\lim }\limits_{x \to 1} f(x) = 3[/imath] BUT [imath]\bf f(1)=4[/imath]
[imath][/imath]
Correct. But if you know the function is continuous, then you can just use f(a).
You can substitute, under the right conditions, namely when you know the function is continuous.
What he said you can't do is "simply" substitute -- that is, do so blindly without checking whether it is valid.
The error in your statement "My understanding is that limits do not equal the value of a function" was the omission of a single word: "My understanding is that limits do not necessarily equal the value of a function."