Q1Mathematics I

Question

Find the limit of the sequence $⟨ x_{n} ⟩$ , where

x_{n} = \frac{5 n - 3}{7 n + 8}

Answer 1

To systematically evaluate the mathematical limit of this rational sequence as $n$ approaches infinity, we must rigorously divide both the numerator and the denominator by the absolutely highest power of $n$ present in the entire expression.

n \to \infty lim \frac{5 n - 3}{7 n + 8} = n \to \infty lim \frac{n ( 5 - 3/ n )}{n ( 7 + 8/ n )}

By canceling out the common $n$ term, we are left with a simplified fraction. As $n$ grows infinitely large, any constant divided by $n$ explicitly approaches zero. Thus, the limit perfectly converges:

n \to \infty lim \frac{5 - 3/ n}{7 + 8/ n} = \frac{5 - 0}{7 + 0} = \frac{5}{7}

Q1Mathematics I

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