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A similar question appeared here Sequence version of L'Hospital's Rule for example, but something is still unclear for me.. If I only had L'Hositals rule for sequences in lecture and don't know the Stolz-Cesaro theorem, is it allowed to apply L'Hosital on a sequence (for example if I want to calculate the limit of $\frac{3n+2}{n+2}$, $n$ goes to infinity)? And if not, what if the problem? Regards

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Flap
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  • One problem is that the expression you have might not be convergent if you put in real numbers. For instance $a_n = \sin(2\pi n)$ is constantly equal to $0$, but $f(x) = \sin(2\pi x)$ does not converge. – Arthur Apr 19 '15 at 11:30
  • Thank you. @elDin0 L'hospital's rule is valid for infinity/infinity too I think – Flap Apr 19 '15 at 11:38
  • What do you mean by "L'Hospital for sequences" to begin with? How do you take the derivative of a sequence? – Hans Lundmark Apr 19 '15 at 11:43
  • @HansLundmark For sequences, you could just use the difference function. Say you have $a_n = \frac{b_n}{c_n}$, and $b_n, c_n \to \infty$. Then you can look at $\frac{b_n-b_{n-1}}{c_n - c_{n-1}}$, and it would work, at least in simple cases. Also note that polynomials have a derivative, regardless of what kind of space the indeterminates live in. Lastly, if you have an expression for the sequence that works for most real numbers, you can make it into a function and use continuous analysis on that function to find the limit. – Arthur Apr 19 '15 at 11:48
  • @Arthur: Well, that is the Stolz–Cesàro theorem. The OP seems to aks about something called "L'Hospital for sequences" covered in lectures, which is not(?) the S–C theorem... – Hans Lundmark Apr 19 '15 at 13:41
  • @Arthur but why is this realy a counterexample? this is not a typical sequence for L'hospital, or am I wrong? – Flap Apr 21 '15 at 06:22

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If $f,g:[1,\infty]\longrightarrow\Bbb R$ are s.t. $$\exists\lim_{x\to\infty}\frac{f(x)}{g(x)} = L,$$ then $$\exists\lim_{n\to\infty}\frac{f(n)}{g(n)} = L.$$ (Quick and dirty proof: $\exists\lim$ in the whole space $\implies\exists\lim$ in the subset)

And now, you can use L'Hôpital in the first limit.

Martín-Blas Pérez Pinilla
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L'Hosital rule is based on the idea of obtaining a 0/0 or infinity/infinity and then you can use the derivative of the numerator and the derivative of the denominator. However, in this case you only need to look at the coefficients of the leading terms. The limit would = 3.

DHouse
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  • Yes, I know how to do it without L'Hospital. For $\lim_{n\to\infty}\frac{3n+2}{n+2}$, we are in the situation $\frac{\infty}{\infty}$, – Flap Apr 19 '15 at 11:41
  • I'm not certain if this really addresses the OPs question. The question is more about whether or not l'Hopital's rule can be applied to sequences (and under what conditions it can be applied), not so much what is the limit of that particular sequence. – TravisJ Apr 19 '15 at 12:06