General questions about limits of sequences

[JulienB]

Hi everybody! I'm currently preparing a math exam, and I'd like to clear up a few points I find obscure about limits of sequences, my goal being to more or less determine a method to solve them quickly during the exam. Hopefully someone can help me here :) I'll number the questions so that it's easier to answer:

1. When I encounter an indeterminate form, the first step I do is always to remove the constants (if any) out of the limit and then rewrite it in the form 0/0 or ∞/∞ (to use L'Hopital), or to apply e or ln on both sides (if the "x" is in the power for example). Is it always correct to write e^{lim a_n} = e^L and ln(lim a_n) = ln(L) or are there exceptions?
For example, let's say that the sequence goes towards -∞: that's a place where e^x never goes, right? Does that mean I should rather use ln than e when facing the possibility of a choice?
(Well maybe there's always a (-1) somewhere in such cases, I don't really have an example right now)

Any thing to add about this first step? What do you guys do?

2. Yesterday I was trying to find the limit for (1/(x-1) - 1/(lnx)) for x→1. I rewrote it in the form 0/0 and used L'Hopital, which gave me an indeterminate form again. I thought I failed, so I looked up Wolfram Alpha and it said to perform the L'Hopital rule twice! Is that a common thing to do, or was there an easier method to solve that?

3. Are there other efficient methods to find limits of sequences? I'm especially fond of those you can apply for "special cases" of sequences (like for example e when you see x in the power).Thank you very much in advance for your answers, I'm sure there's already enough matter to discuss in those 3 questions.Julien.

 

[member 587159]

Hi everybody! I'm currently preparing a math exam, and I'd like to clear up a few points I find obscure about limits of sequences, my goal being to more or less determine a method to solve them quickly during the exam. Hopefully someone can help me here :) I'll number the questions so that it's easier to answer:

1. When I encounter an indeterminate form, the first step I do is always to remove the constants (if any) out of the limit and then rewrite it in the form 0/0 or ∞/∞ (to use L'Hopital), or to apply e or ln on both sides (if the "x" is in the power for example). Is it always correct to write e^{lim a_n} = e^L and ln(lim a_n) = ln(L) or are there exceptions?
For example, let's say that the sequence goes towards -∞: that's a place where e^x never goes, right? Does that mean I should rather use ln than e when facing the possibility of a choice?
(Well maybe there's always a (-1) somewhere in such cases, I don't really have an example right now)

Any thing to add about this first step? What do you guys do?

2. Yesterday I was trying to find the limit for (1/(x-1) - 1/(lnx)) for x→1. I rewrote it in the form 0/0 and used L'Hopital, which gave me an indeterminate form again. I thought I failed, so I looked up Wolfram Alpha and it said to perform the L'Hopital rule twice! Is that a common thing to do, or was there an easier method to solve that?

3. Are there other efficient methods to find limits of sequences? I'm especially fond of those you can apply for "special cases" of sequences (like for example e when you see x in the power).Thank you very much in advance for your answers, I'm sure there's already enough matter to discuss in those 3 questions.Julien.

With good practice, you can determine limits on first sight. Also, it is normal you sometimes have to use l'Hospital twice.