Suppose
\(f(x)=\dfrac{4x^7-2x^2-9x+3}{7x^3-5x^2-9x+1}\) and we want to find its end behavior. Our first step is to isolate the leading term of top and bottom of the fraction and then simplify it:
\begin{equation*}
\frac{4x^7}{7x^3} = \frac{4x^4}{7}
\end{equation*}
Since there are leftover
\(x\)βs in the numerator, we need to ask: "What happens when we plug in huge numbers for
\(x\text{?}\)" In this case, we are left with something weβve seen before: a polynomial! The leading term is
\(\frac{4}{7}\text{,}\) which is positive, and the degree is
\(4\text{,}\) which is even. So, the end behavior is
-
As \(x\rightarrow \infty\text{,}\) \(y\rightarrow \infty\text{.}\)
-
As \(x\rightarrow -\infty\text{,}\) \(y\rightarrow \infty\text{.}\)