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

1. May 9, 2013

### Kenji Liew

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. May 9, 2013

### 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.

3. May 9, 2013

### Kenji Liew

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.