The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. \(\displaystyle\lim_{{x} \rightarrow \infty}\frac{{{x}−{3}}}{{\sqrt{{4}{x}^{{2}}+{25}}}}\)

Determine \(\displaystyle\lim_{x→∞}{f{{\left({x}\right)}}}\) and \(\displaystyle\lim_{x→−∞}{f{{\left({x}\right)}}}\) for the following rational functions. Then give the horizontal asymptote of f (if any). \(\displaystyle{f{{\left({x}\right)}}}=\frac{{{40}{x}^{{5}}+{x}^{{2}}}}{{{16}{x}^{{4}}−{2}{x}}}\)

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. \(\lim_{x\to\infty}((x^{−1})+(x^{−4}))/((x^{−2})−(x^{−3}))\)