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

[ElectroPhysics]

1) can we say limit of a function f(x) such that limit_{x\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 limit_{x\rightarrowa} f(x) = slope

3) can we say that if left hand limit = right hand limit and limit_{x\rightarrowa}f(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

 

[Pere Callahan]

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.

 

[ElectroPhysics]

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.

 

[Pere Callahan]

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.

 

[ElectroPhysics]

what about these three points

3) can we say that if left hand limit = right hand limit and limit_{x\rightarrowa}f(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

 

[Pere Callahan]

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.

 

[ElectroPhysics]

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.

 

[HallsofIvy]

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.

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.

 

[HallsofIvy]

1) can we say limit of a function f(x) such that limit_{x\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 limit_{x\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 "limit_{x\rightarrowa} f(x) = slope"

3) can we say that if left hand limit = right hand limit and limit_{x\rightarrowa}f(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.