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Physical significance of limit

  1. Nov 22, 2008 #1
    1) can we say limit of a function f(x) such that limitx[tex]\rightarrow[/tex]a 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[tex]\rightarrow[/tex]a f(x) = slope

    3) can we say that if left hand limit = right hand limit and limitx[tex]\rightarrow[/tex]af(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
  2. jcsd
  3. Nov 22, 2008 #2
    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.
  4. Nov 24, 2008 #3
    Does it says that limit is just a process to find corresponding f(x) values for some x values.
  5. Nov 24, 2008 #4
    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.
  6. Nov 24, 2008 #5
    what about these three points

  7. Nov 24, 2008 #6
    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.
  8. Nov 24, 2008 #7

    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.
  9. Nov 24, 2008 #8


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    No, it definitely is not! Too many beginning students get the impression that "[itex]\lim_{x\rightarrow a} f(x)[/itex]" 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.
  10. Nov 24, 2008 #9


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    No, we can't. As others have said, the derivative is the limit of the "difference quotient" not the limit of the function itself.

    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[tex]\rightarrow[/tex]a f(x) = slope"

    Yes, that is the definition of "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.

    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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