Question

find the end behavior of the following rational functions. (g(x)=\frac{(2x + 5)(x - 4)}{x+1})

Explanation:

Step1: Expand the numerator

First, expand \((2x + 5)(x - 4)\) using FOIL method. \((2x+5)(x - 4)=2x\times x-2x\times4+5\times x - 5\times4=2x^{2}-8x + 5x-20=2x^{2}-3x - 20\). So \(g(x)=\frac{2x^{2}-3x - 20}{x + 1}\).

Step2: Analyze end - behavior as \(x\to\pm\infty\)

For a rational function \(y=\frac{f(x)}{h(x)}\) where \(f(x)=a_nx^n+\cdots+a_0\) and \(h(x)=b_mx^m+\cdots + b_0\), the end - behavior is determined by the leading terms of the numerator and denominator. Here, the leading term of the numerator is \(2x^{2}\) and the leading term of the denominator is \(x\). As \(x\to\pm\infty\), \(g(x)\approx\frac{2x^{2}}{x}=2x\). So \(\lim_{x\to\infty}g(x)=\infty\) and \(\lim_{x\to-\infty}g(x)=-\infty\).

Answer:

\(\lim_{x\to\infty}g(x)=\infty,\lim_{x\to-\infty}g(x)=-\infty\)