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)