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Why L' Hospital's rule cannot apply directly for sequence

  1. May 9, 2013 #1
    1. The problem statement, all variables and given/known data

    Well, this thread is purposely to clarify my question on the use of L'Hospital's rule for sequence.
    As I have read from the calculus book, the sequence can be defined such that f:N → ℝ with function f(n)= an where n inside natural numbers,N.

    So, we cannot apply L' Hospital rule directly is because the function above is discontinuous on natural numbers and then not differentiable?

    2. Relevant equations

    When we plot the graph f(n) versus n, we don't have a continuous graph. Is this mean discontinuous?


    3. The attempt at a solution
     
  2. jcsd
  3. May 9, 2013 #2

    Mark44

    Staff: Mentor

    The graph of a sequence represents a discrete (separated) set of points. The graph is discontinuous, which is a synonym for not continuous. A function that isn't continuous is also not differentiable. However, you can often extend the sequence to a function that is continuous and differentiable.
     
  4. May 9, 2013 #3
    Thanks for your prompt reply. I think I can understand after looking at your explanation.
     
  5. May 9, 2013 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    If you have, for example, [itex]\lim_{n\to\infty}\frac{3n^2+ 3n-1}{5n^2- 5n+ 2}[/itex], then you can use L'hopital's rule. [itex]\lim_{x\to\infty} f(x)= a[/itex] if and only if [itex]\lim_{n\to\infty} f(x_n)= a[/itex] for every sequence [itex]x_n[/itex] such that [itex]x_n[/itex] that goes to infinity.

    So if you have a sequence that can be written as a continous function (by allowing n to have any real number as a value rather than only integer values) then you can use L'Hopital's rule. On the other hand, things that cannot be extended to all real numbers, such as "n!", cannot be (easily) extended to real variables, cannot be done that way.

    (n! can be extended to the "gamma function" but trying to apply L'Hopital's rule to that is likely to be exteremely difficult.)
     
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