2.1.8 Calculation of limits

Problem: 1. Based on the definition of the limit of a sequence, prove that the limit of the sequence \( \left\{x_{n}\right\}=\left\{\frac{7 \cdot 4^{n}+3}{5 \cdot 4^{n}+4}\right\} \) is equal to the number \( a=\frac{7}{5} \), or disprove this statement. 2. If the statement is true, then for \( \varepsilon=\frac{1}{10000} \) find an \( N \), such that for all \( n>N \) the inequality \( \left|x_{n}-a\right|<\varepsilon \) holds.