Final answer:
To find the limit of the sequence b₍ = (ln(n+1) - ln(n)) / (ln(n) ⋅ ln(n+1)), it simplifies to 1 / (ln(n) ⋅ ln(n+1)). As n approaches infinity, both ln(n) and ln(n+1) approach infinity, therefore, the limit of the sequence is 0.
Step-by-step explanation:
To find the limit of the sequence b₍ = (ln(n+1) - ln(n)) / (ln(n) ⋅ ln(n+1)), we can simplify it by using the properties of logarithms. Specifically, the property that allows us to express the difference of logarithms as the logarithm of a quotient: ℓ log(a) - ℓ log(b) = ℓ log(a/b). This simplifies the original sequence to b₍ = 1 / (ln(n) ⋅ ln(n+1)).
As n approaches infinity, both ln(n) and ln(n+1) also approach infinity, so their product does as well. Therefore, we get:
Limit of b₍ as n approaches infinity = 0.
This result is justified by the knowledge that the natural log function grows without bounds, but at a rate slower than linear functions, polynomials, or exponentials. So as n gets large, the denominator of our sequence gets infinitely large, meaning the overall sequence approaches zero