# Rational functions questions and answers

Recent questions in Rational functions
Rational functions

### Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $$\displaystyle\int{\frac{{{\cos{{x}}}}}{{{\sin{{x}}}{\left({1}-{\sin{{x}}}\right)}}}}{\left.{d}{x}\right.}$$

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. $$\displaystyle\lim_{{x}→∞}\frac{{{x}−{3}}}{{√{4}{x}^{{2}}+{25}}}$$

Rational functions

### Determine $$\displaystyle\lim_{{{x}\rightarrow\infty}}{f{{\left({x}\right)}}}$$ and $$\displaystyle\lim_{{{x}\rightarrow-\infty}}{f{{\left({x}\right)}}}$$ for the following rational functions. Then give the horizontal asymptote of f (if any). $$\displaystyle{f{{\left({x}\right)}}}={\frac{{-{x}^{{{3}}}+{1}}}{{{2}{x}+{8}}}}$$

Rational functions

### For the following exercises, find the domain of the rational functions. $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}^{{{2}}}+{4}}}{{{x}^{{{2}}}-{2}{x}-{8}}}}$$

Rational functions

### Investigate asymptotes of rational functions. Copy and complete the table. Determine the horizontal asymptote of each function algebraically. Function Horizontal Asymptote $$\displaystyle{f{{\left({x}\right)}}}=\frac{{{x}^{{2}}-{5}{x}{4}}}{{{x}^{{3}}+{2}}}$$ $$\displaystyle{h}{\left({x}\right)}=\frac{{{x}^{{3}}-{3}{x}^{{2}}+{4}{x}-{12}}}{{{x}^{{4}}-{4}}}$$ $$\displaystyle{g{{\left({x}\right)}}}=\frac{{{x}^{{4}}-{1}}}{{{x}^{{5}}+{3}}}$$

Rational functions

### (a) rewrite the sum as a rational function S(n), (b) use S(n) to complete the table, and (c) find $$\displaystyle\lim_{{n}→∞}{S}{\left({n}\right)}$$ $$S(n)=100, 101, 102, 103, 104$$ $$\displaystyle{\sum_{{{i}={1}}}^{{n}}}{\frac{{{3}}}{{{n}^{{{3}}}}}}{\left({1}+{i}^{{{2}}}\right)}$$

Rational functions

### Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $$\displaystyle\int{\frac{{{1}+\epsilon^{{{x}}}}}{{{1}-\epsilon^{{{x}}}}}}{\left.{d}{x}\right.}$$

Rational functions

### Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $$\displaystyle\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{x}}}+\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}}}}$$

Rational functions

### Determine $$\displaystyle\lim_{{{x}\rightarrow\infty}}{f{{\left({x}\right)}}}$$ and $$\displaystyle\lim_{{{x}\rightarrow-\infty}}{f{{\left({x}\right)}}}\rbrace$$ for the following rational functions. Then give the horizontal asymptote of f (if any). $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}{4}+{7}}}{{{x}{5}+{x}^{{{2}}}-{x}}}}−{x}$$

Rational functions

### Graph each of the following rational functions. State the equations of the asymptotes. a) $$\displaystyle{j}{\left({x}\right)}=-{\frac{{{6}}}{{{x}+{4}}}}+{7}$$ b) $$\displaystyle{k}{\left({x}\right)}={\frac{{{x}+{6}}}{{{2}{x}-{4}}}}$$

Rational functions

### Consider the following sets of rational functions. $$\displaystyle{a}{f{{\left({x}\right)}}}={\frac{{{a}}}{{{x}}}}$$ for a={−2,−1,−0.5,0.5,2,4}, $$\displaystyle{h}{\left({x}-{c}\right)}={\frac{{{1}}}{{{x}-{c}}}}$$​ for c=[−4,−2,−0.5,0.5,2,4], h(x−c)=1x−c for c=[−4,−2,−0.5,0.5,2,4], g(bx)=1bx for b={−2,−1,−0.5,0.5,2,4} for b={−2,−1,−0.5,0.5,2,4}, $$\displaystyle{k}{\left({x}\right)}+{d}={\frac{{{1}}}{{{x}}}}+{d}$$ for d={−4,−2,−0.5,0.5,2,4} for d={−4,−2,−0.5,0.5,2,4}. a. Graph each set of functions on a graphing calculator.

Rational functions

### $$\displaystyle\lim_{{{x}\rightarrow-\infty}}{\frac{{\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}-\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}}}{{\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}+\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}}}}$$

Rational functions

### Let F be the field of rational functions described in the Example. (a) Show that the ordering given there satisfies the order axioms O1, O2, and O3. (b) Write the following polynomials in order of increasing size: $$\displaystyle{x}^{{{2}}},−{x}^{{{3}}},{5},{x}+{2},{3}−{x}$$. (c) Write the following functions in order of increasing size: $$\displaystyle{\frac{{{x}^{{{2}}}+{2}}}{{{x}-{1}}}},{\frac{{{x}^{{{2}}}-{2}}}{{{x}+{1}}}},{\frac{{{x}+{1}}}{{{x}^{{{2}}}-{2}}}},{\frac{{{x}+{2}}}{{{x}^{{{2}}}-{1}}}}$$.

Rational functions

### Consider the following sets of rational functions. $$\displaystyle{a}{f{{\left({x}\right)}}}={\frac{{{a}}}{{{x}}}}$$ for a={−2,−1,−0.5,0.5,2,4}, $$\displaystyle{h}{\left({x}-{c}\right)}={\frac{{{1}}}{{{x}-{c}}}}$$​ for c=[−4,−2,−0.5,0.5,2,4], h(x−c)=1x−c for c=[−4,−2,−0.5,0.5,2,4], g(bx)=1bx for b={−2,−1,−0.5,0.5,2,4} for b={−2,−1,−0.5,0.5,2,4}, $$\displaystyle{k}{\left({x}\right)}+{d}={\frac{{{1}}}{{{x}}}}+{d}$$ for d={−4,−2,−0.5,0.5,2,4} for d={−4,−2,−0.5,0.5,2,4}. c. Choose two functions from any set. Find the slope between consecutive points on the graphs.

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. $$\displaystyle\lim_{{{x}\rightarrow-\infty}}\sqrt{{{\frac{{{x}^{{{2}}}-{5}{x}}}{{{x}^{{{3}}}+{x}-{2}}}}}}$$

Rational functions

### Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $$\displaystyle\int{\frac{{\epsilon^{{{x}}}}}{{{36}-\epsilon^{{{2}{x}}}}}}{\left.{d}{x}\right.}$$. (Give the exact answer and the decimal equivalent. Round to five decimal places.)

Rational functions

### Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $$\displaystyle\int{\frac{{{1}-\sqrt{{{x}}}}}{{{1}+\sqrt{{{x}}}}}}{\left.{d}{x}\right.}$$

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. limx→∞((x^−1)+(x^−4))/((x^−2)−(x^−3))

Rational functions