Is the Limit of a Function the Slope Around a Point?

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Discussion Overview

The discussion revolves around the relationship between limits and slopes in the context of calculus, specifically addressing whether the limit of a function at a point can be equated to the slope of the function at that point. Participants explore concepts related to continuity, derivatives, and the definitions of limits and slopes.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the limit of a function at a point can be interpreted as the slope of the function around that point, while others strongly disagree, stating that the slope is defined by a limit process related to the derivative, not the limit of the function itself.
  • There is a discussion about the necessity of a specific point "a" when discussing limits and slopes, with some arguing that limits can exist independently of the function being defined at that point.
  • Participants discuss the definition of continuity, agreeing that if the left-hand limit equals the right-hand limit and the limit equals the function value at that point, then the function is continuous.
  • Some participants clarify that while a derivative exists at a point, it is not simply the slope of a continuous function, as continuity does not guarantee differentiability.
  • There is a contention regarding the need for a specific point when finding derivatives, with some asserting that derivatives are defined at specific points, while others suggest that derivatives can be discussed more generally without reference to a specific point.

Areas of Agreement / Disagreement

Participants express disagreement on the relationship between limits and slopes, with no consensus reached on whether the limit of a function can be equated to the slope at a point. There is some agreement on definitions of continuity, but differing views on the implications of derivatives and their relationship to slopes.

Contextual Notes

Limitations include varying interpretations of the definitions of limits, slopes, and derivatives, as well as the conditions under which these concepts apply. The discussion highlights the complexity of these mathematical concepts and the nuances involved in their definitions.

ElectroPhysics
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1) can we say limit of a function f(x) such that limitx\rightarrowa f(x) gives the slope of this function around this point x = a.

2) can we say that to find slope i.e. limit of a function f(x) we always need a point "a" such that limitx\rightarrowa f(x) = slope

3) can we say that if left hand limit = right hand limit and limitx\rightarrowaf(x) = f(a) then it is a continuous function.

4) can we say that if a derivative exist then it is just the slope of a continuous function.

5) can we say that in order to find derivative we normally don't need above mentioned point x = a
 
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ElectroPhysics said:
1) can we say limit of a function f(x) such that \lim_{x\to a}f(x) gives the slope of this function around this point x = a.
No! The slope, or derivative, is defined by a limit process, but the limit of a function itself has nothing to do with its slope.
 
Pere Callahan said:
No! The slope, or derivative, is defined by a limit process, but the limit of a function itself has nothing to do with its slope.

Does it says that limit is just a process to find corresponding f(x) values for some x values.
 
Yes, basically, and it might also work to find the limit of a function f at some point x, even though f is not defined at x.
 
what about these three points


ElectroPhysics said:
3) can we say that if left hand limit = right hand limit and limitx\rightarrowaf(x) = f(a) then it is a continuous function.

4) can we say that if a derivative exist then it is just the slope of a continuous function.

5) can we say that in order to find derivative we normally don't need above mentioned point x = a
 
ElectroPhysics said:
what about these three points

3) yes, that is one definition of continuity.

4) Yes.

5) What do you mean by "need". In order for the derivative to exist at some point, the function must be at least defined and continuous there.
 
Pere Callahan said:
5) What do you mean by "need". In order for the derivative to exist at some point, the function must be at least defined and continuous there.


suppose y = f(x) = x3 then derivative of f(x) is just 3x2
i.e. without knowing domain of f(x) we have found the derivative.
 
ElectroPhysics said:
Does it says that limit is just a process to find corresponding f(x) values for some x values.

Pere Callahan said:
Yes, basically, and it might also work to find the limit of a function f at some point x, even though f is not defined at x.
No, it definitely is not! Too many beginning students get the impression that "\lim_{x\rightarrow a} f(x)" is just a complicated way of talking about f(a) but that is certainly not true. An example I like to use is
f(x)= x2 if x< -0.00001
f(x)= x+ 10000 if -0.0001<= x< 0
f(0)= -100
f(x)= 10000- x2 if 0< x< 0.00001
f(x)= x2 if x> 0.00001

The limit of f(x), as x goes to 0, is, of course, 10000.
 
ElectroPhysics said:
1) can we say limit of a function f(x) such that limitx\rightarrowa f(x) gives the slope of this function around this point x = a.
No, we can't. As others have said, the derivative is the limit of the "difference quotient" not the limit of the function itself.

2) can we say that to find slope i.e. limit of a function f(x) we always need a point "a" such that limitx\rightarrowa f(x) = slope
Again, "slope" and "limit of a function" are not the same. In fact, strictly speaking, only straight lines have "slope". It is true that the derivative is the slope of the tangent line.
It is NOT true that "limitx\rightarrowa f(x) = slope"

3) can we say that if left hand limit = right hand limit and limitx\rightarrowaf(x) = f(a) then it is a continuous function.
Yes, that is the definition of "continuous function".

4) can we say that if a derivative exist then it is just the slope of a continuous function.
Bad wording. Again, only straight lines have "slope". Also, continuous function may not be differentiable. Finally, it makes no sense to talk about a 'derivative' without saying derivative of a specific function. It is true that if a function has a derivative at a specific point then it is (by definition) differentiable at that point, there is a tangent line to its graph at that point, and the derivative at a point is the slope of the tangent line.

5) can we say that in order to find derivative we normally don't need above mentioned point x = a
The derivative is by definition at a specific point, whether you call it "a" or not. you can then talk about the derivative function: the function that gives the derivative at each x. But when you write d(x2/dx= 2x, you are still talking about the derivative at individual values of x.

Oh, and finally, none of this has anything at all to do with a "physical" meaning of the derivative.
 

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