There are vertical asymptotes at x=2 and x=-8/3 There are no extrema. There is a point of inflection at (1,0) The graph is decreasing on intervals (-inf, -8/3), (-8/3, 2), and (2, inf) No one viewing window is perfect. If you zoom in you can see x intercept and point of inflection. You would need to zoom out to see horizontal asymptotes. If x = 890, then y= 0.6676664, which is within 0.001 of 2/3 or 0.6666…..
Determine (finite) limits at infinity Determine horizontal asymptotes, if any, of the graph Determine infinite limits at infinity
What is the “end behavior” of a function on an infinite interval? You could look at graph, or look at table of values Limits at infinity occur as x increases or decreases without bound x - ∞← -10 -1 0 1 10 ->∞ f(x) 3 ← 2.97 1.5 0 1.5 2.97 -> 3
As x increases to infinity, the graph gets trapped between horizontal lines y=L + ε and y=L – ε . Notice that the graph can CROSS a horizontal asymptote.
Use a graphing utility to graph Describe all the important features of the graph. Can you find a single viewing window that shows all these features clearly? Explain. What are the horizontal asymptotes of the graph? How far to the right do you have to move on the graph so that the graph is within 0.001 unit of its horizontal asyptote? Explain.
Limits at infinity have many of same properties as those we saw in Section 1.3 For example,
Find the limit: If we look at the numerator and denominator separately, each would be going to infinity, so we would get which is an indeterminate form To resolve this issue, divide both the numerator and denominator by the highest power of x in the denominator , which in this case is x So line y=2 is horiz. asymptote to the right. If you take the limit as xapproaches -∞, it also approaches y = 2
One way to test the reasonableness of a limit is to evaluate f(x) at a few large values of x. You also could graph f(x) and the horizontal asymptote you found and see if the graph of f(x) looks like it moves closer and closer to the horizontal asymptote.
In each case, trying to evaluate the limit produces indeterminate form ∞/ ∞
Rational functions approach the SAME horizontal asymptote as you go to infinity to the right or left. Functions that are not rational might have different behaviors at each end. Look at a. and b.
In Section 1.3 you saw the squeeze theorem and how it works as x approaches 0 with trig functions. This theorem can also work when x approaches infinity. Ex. 5 p. 203 Limits involving trig functions at infinity Find each limit: Solution: as x approaches infinity, the graph oscillates between 1 and -1. So no limit exists. Solution: Because -1 ≤ sin x ≤ 1, for x>0, Limits at outside equal 0, so the limit in between has to equal 0 from squeeze theorem.
Determining whether a function has an infinite limit at infinity is useful for analyzing the “end behavior” of its graph.
Find and Solution: a. As x increases without bounds, so does y. So limit equals infinity. b. As x decreases without bounds, so does y. So limit equals negative infinity.
Ex 8 pg. 204 Using long division to work with “top heavy” rational functions. Find: These statements can be interpreted as saying that as x approaches ∞, the function behaves just like g(x) = 2x – 6. In section 3.6 we will describe that graphically as a slant asymptote.