Dominance and Limits to Infinity

When we consider , we are really considering which term in the function expression "dominates" (has the most effect on) the function as becomes large in magnitude. For example, in the function , the term has a much larger magnitude than when is a large negative number. But when is large and positive, is much larger than . So we say that is an End Behavior Model to the left (negative), and is an End Behavior Model to the right (positive). The graph below starts with a vew near the origin. At these small magnitudes of , it can be difficult to make any predictions about what happens further away from the origin. See if you can find the Left and Right End Behavior Models (EBM) for the function, type them in, then zoom out using the two sliders to test your answer. If you are correct, the red and green EBMs will merge with the black function, better as moves away from the origin.