Submit quiz Follow the steps for graphing a rational function to graph the function $R(x)=\frac{x+2}{x(x+11)}$. Determine the behavior of the graph on either side of any vertical asymptotes, if one exists. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. It approaches $\infty$ on one side of the asymptote(s) at $x=\square$ and $-\infty$ on the other. It approaches either $\infty$ or $-\infty$ on both sides of the asymptote(s) at $x=\square$. (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) B. It approaches either $\infty$ or $-\infty$ on both sides of the asymptote(s) at $x=\square$. (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) C. It approaches $\infty$ on one side of the asymptote(s) at $x=\square$ and $-\infty$ on the other. (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) D. The function has no vertical asymptote. Determine the horizontal asymptote(s), if one exists. Select the correct choice below and, if necessary, fill in the
Solution
Step 1 ：Set the denominator equal to zero and solve for x: \(x(x+11) = 0\). This gives us two solutions: \(x = 0\) and \(x = -11\). These are the vertical asymptotes.
Step 2 ：Plug in values of x that are slightly less than and slightly greater than the asymptotes into the function and see what happens. For \(x = 0\), the graph approaches infinity on the left side and negative infinity on the right side. For \(x = -11\), the graph approaches negative infinity on the left side and infinity on the right side.
Step 3 ：The horizontal asymptote of a rational function can be found by looking at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). In this case, the degree of the numerator \((x+2)\) is 1 and the degree of the denominator \((x(x+11))\) is 2. So, the horizontal asymptote is \(y = 0\).
