Questions: Does the sequence (4n-3)/(3n+5) converge? If so, what is the limit?

Solution Steps

We are given the sequence defined by the expression

\[ a_n = \frac{4n - 3}{3n + 5}. \]

To determine the convergence of the sequence as \( n \) approaches infinity, we analyze the limit:

\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{4n - 3}{3n + 5}. \]

As \( n \) approaches infinity, we focus on the leading terms in the numerator and denominator:

\[ \lim_{n \to \infty} \frac{4n - 3}{3n + 5} = \lim_{n \to \infty} \frac{4n}{3n} = \frac{4}{3}. \]

Final Answer