My question might sound a bit strange, maybe not, but it is of great importance for me that someone answers it thoroughly...
Let's say we have the following sequence >>
a(n) := n / [(n^2) - 4]
if i were to calculate the lim n --> infinite i would 1 as the result.
but if i were to divide both the numerator and the denominator with n the sequence would change to a(n) := 1 / [n - (4/n)] and then the lim n --> infinite would be 0. Now the sequence is not convergent 1 but a nullsequence.
Why is convergent 0 correct and convergent 1 false? As correct answer i was told 0, but why? why not 1? must i always try and divide + simplify?
2 >> and is there a clear way to simplify before executing the lim? For example, every variable in the numerator must be of power 1 or something?
I hope my question is clear for you and you can help me. thank you
for an nth term of a sequence in the form of a quotient ... when the degree of the denominator > degree of the numerator, the limit of the sequence as n tends to infinity is zero.
you should already be familiar with this concept after studying the end-behavior of rational functions in prior algebra / precalculus courses.
and if the degree of the numerator > degree of the denominator then the limit with an n that tends to infinity is infinite?
unfortunately i never had the opportunity to learn, at least from the school or university books up until now i never did anything like that before..