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

[Kenji Liew]

Homework Statement

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)= a_n 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?

Homework Equations

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

The Attempt at a Solution

 

[Mark44]

Homework Statement

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)= a_n 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?

Homework Equations

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

The Attempt at a Solution

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.

 

[Kenji Liew]

Thanks for your prompt reply. I think I can understand after looking at your explanation.

 

[HallsofIvy]

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

So if you have a sequence that can be written as a continuous 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.)