\( \def\dfrac#1#2{\displaystyle\frac{#1}{#2}} \def\solve#1{\begin{array}{rcl}#1\end{array} } \)

Home / 06 Rational Functions / 06 End Behavior Of Rationals

Example: Determine the end behavior of the rational functions listed below.

  1. \(f(x)=\dfrac{3x^4-x^2+2}{x+4}\)


  2. \(g(x)=\dfrac{-7x^5-x^3+2x}{x^7+5x^4-3}\)


  3. \(k(t)=\dfrac{-t^2+2t+5t^3}{t^3-27}\)

Solution

  1. End Behavior determined by: \[\dfrac{3x^4}{{x}} =3x^3\]This is an odd degree polynomial with positive leading coefficient, thus, the end behavior is \[ \solve{ y\rightarrow-\infty &\text{ as } &x\rightarrow -\infty\\ y\rightarrow \infty &\text{ as} &x\rightarrow \infty } \]
  2. End Behavior determined by: \[\dfrac{-7x^5}{x^7} =\dfrac{-7}{x^2}\] this confirms that the denominator dominates the rational function, thus, the end behavior is a horizontal asympotote at \(y=0\).
  3. End Behavior determined by: \[\dfrac{5t^3}{t^3} =5\] this confirms that the degree of the numerator and denominator are the same and that the end behavior is a horizontal asymptote at \(y=5\)