\( \lim _ { n \rightarrow \infty } \frac { 4 x ^ { 2 } + x - 1 } { 8 m ^ { 2 } + x } \)

To evaluate the limit as \( n \) approaches infinity for the expression \( \frac{4x^2 + x - 1}{8m^2 + x} \), we first observe that the variables \( n \) and \( m \) aren’t directly linked in the expression provided. Assuming \( x \) is the variable approaching a limit, we can simplify this. If \( x \) remains finite, the degrees of the polynomial in the numerator and denominator affect the limit. The highest degree term in the numerator is \( 4x^2 \), and in the denominator, we have \( 8m^2 \). As \( n \) approaches infinity, if we let \( m \) also approach infinity, then the limit can be evaluated by analyzing the leading terms. If we keep \( m^2 \) as significantly larger than \( x \) as \( n \) increases, the limit simplifies to: \[ \lim _{n \rightarrow \infty} \frac{4x^2 + x - 1}{8m^2} = 0. \] This is because the numerator is a polynomial of finite degree while the denominator increases without bound. However, if we assume \( m \) is a fixed constant or not tending to infinity, then the limit does not exist as \( n \) becomes asymptotic, suggesting unraveling additional context is necessary for a clearer interpretation.