The general formula for evaluating the limit of a sequence is:
lim(n→∞) a_n = L, where n represents the term number and L is the limit value.
A sequence is said to be convergent if its limit exists and is a finite value. This means that as the term number approaches infinity, the terms of the sequence get closer and closer to a specific value. A sequence is divergent if its limit does not exist or if it approaches infinity or negative infinity.
The limit of the sequence 3^n/(3^n + 2^n) is 1. This can be determined by dividing both the numerator and denominator by the highest power of n, in this case 3^n. This results in 3^n/3^n + 2^n/3^n. As n approaches infinity, 3^n/3^n becomes equal to 1 and 2^n/3^n approaches 0. Therefore, the limit is 1.
No, L'Hopital's rule is only applicable for evaluating the limit of a function, not a sequence. The limit of a sequence is evaluated by finding the behavior of the terms as n approaches infinity, not by taking the derivative of the function.
To prove that the limit of a sequence is equal to a specific value L, you must show that for any value ε > 0, there exists an N such that for all n > N, |a_n - L| < ε. This means that the terms of the sequence get arbitrarily close to L as n approaches infinity.