Question

which two limits below indicate the end - behavior of the rational function above?
lim_{x->\infty}f(x)=-\infty; lim_{x->-\infty}f(x)=-\infty
lim_{x->\infty}f(x)=-\infty; lim_{x->-\infty}f(x)=\infty
lim_{x->\infty}f(x)=\infty; lim_{x->-\infty}f(x)=-\infty
lim_{x->\infty}f(x)=\infty; lim_{x->-\infty}f(x)=\infty

Explanation:

Step1: Analyze right - hand end - behavior

As \(x\to+\infty\), looking at the graph, the function \(y = f(x)\) approaches positive infinity. So \(\lim_{x\to+\infty}f(x)=\infty\).

Step2: Analyze left - hand end - behavior

As \(x\to-\infty\), looking at the graph, the function \(y = f(x)\) approaches negative infinity. So \(\lim_{x\to-\infty}f(x)=-\infty\).

Answer:

\(\lim_{x\to+\infty}f(x)=\infty;\lim_{x\to-\infty}f(x)=-\infty\)