Recent questions in Rational functions
Rational functions

### 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. $$\displaystyle\lim_{{x} \rightarrow \infty}\frac{{{x}−{3}}}{{\sqrt{{4}{x}^{{2}}+{25}}}}$$

Rational functions

### Determine the domains of the given rational functions. $$\displaystyle\frac{{{x}^{{2}}-{9}}}{{{x}^{{3}}-{x}}}$$

Rational functions

### What formula do we need, in addition to: $$\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({x}^{{n}}\right)}={n}{x}^{{{n}-{1}}}$$ $$\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({c}{u}\right)}={c}{d}\frac{{u}}{{\left.{d}{x}\right.}}$$ $$\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({u}_{{1}}+{u}_{{2}}+\ldots+{u}_{{n}}\right)}={d}\frac{{u}_{{1}}}{{\left.{d}{x}\right.}}+{d}\frac{{u}_{{2}}}{{\left.{d}{x}\right.}}+\ldots+{d}\frac{{u}_{{n}}}{{\left.{d}{x}\right.}}$$ to differentiate rational functions?

Rational functions

### 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). $$\displaystyle{f{{\left({x}\right)}}}=\frac{{{40}{x}^{{5}}+{x}^{{2}}}}{{{16}{x}^{{4}}−{2}{x}}}$$

Rational functions

### In your own words, summarize the guidelines for finding limits at infinity of rational functions.

Rational functions

### Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions f/g, with the degree off ≤≤ the degree ofg (including f = 0).

Rational functions

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