Question 1 Find the horizontal and/or slant asymptote(s) of each function: a) \( f(x)=\frac{5 x^{2}-3 x+2}{-6 x^{2}+8 x^{3}-2} \) b) \( g(x)=\frac{5 x+2}{10 x-5} \)

For the function \( f(x)=\frac{5 x^{2}-3 x+2}{-6 x^{2}+8 x^{3}-2} \), we start by identifying the leading terms of the numerator and denominator. The leading term in the numerator is \( 5x^2 \), and the leading term in the denominator is \( 8x^3 \). Since the degree of the numerator (2) is less than the degree of the denominator (3), the horizontal asymptote is \( y = 0 \). For the function \( g(x)=\frac{5 x+2}{10 x-5} \), we again look for leading terms. The leading term in both the numerator and the denominator is linear: \( 5x \) and \( 10x \), respectively. To find the horizontal asymptote, we divide the coefficients of these leading terms: \( \frac{5}{10} = \frac{1}{2} \). Thus, there is a horizontal asymptote at \( y = \frac{1}{2} \).