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Problem 66 Hard Difficulty

By the \textit{end behavior} of a function we mean the behavior of its values as $ x \to \infty $ and $ x \to -\infty $.
(a) Describe and compare the end behavior of the functions
$$ P(x) = 3x^5 - 5x^3 + 2x \hspace{5mm} Q(x) = 3x^5 $$
by graphing both functions in the viewing rectangles $ [-2, 2] $ by $ [-2, 2] $ and $ [-10, 10] $ by $ [-10,000, 10,000] $.
(b) Two functions are said to have the \textit{same end behavior} if their ratio approaches 1 as $ x \to \infty $. Show that $ P $ and $ Q $ have the same end behavior.

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Video Transcript

this problem number sixty six of these to a calculus eighth edition section two point six By the end behavior of a function we mean the behaviour of its values. As X approaches, infinity and expert is negative. Infinity Party describing compared the end behavior of the functions. P of X is equal to three x to the fifth minus five x cubed plus two x and cue. Vex is equal to three x to the fifth by graphing both functions in the viewing rectangles negative to buy two by native to two, ending a tenner by ten by negative one thousand five one thousand. So these are the functions P and Q. We can go ahead and use any plotting device ah, to dry plot and graph both of these functions next to each other in this first window from negative to two and negative two to two. When we consider the functions are different and they're in behavior seems slightly different, although we can see that they're both increasing as X increases and both decreasing as ex decreases towards negative infinity. So at the moment we say they have similar and behavior as we look at the other being window native. Ten ten by Navy ten thousand one thousand. Here's what that function would look like, and what we see is that the functions are very, very similar. It's very hard to tell the difference between which one is this first function, which in second function wishes P, which is Q. In fact, you could say that they had a very same exactly and behavior seen as his expertise. Infinity. The function approaches infinity for both as expression. They get infinity. The function approaches negative infinity for each function. So for party, we've compared their graphs, and we can say confidently that they're M behavior is very similar. And Part B. We're going to try to prove this. Two functions are said to have the same end behavior get. Their ratio approaches one as expertise. Infinity show that P and Q have the same same and behavior. So we're going to go ahead and take this limit and experts infinity, and we're going to do this for ah, the ratio of the first function. Peep ten Q. So this ratio stone we can and precious limit by they're writing huge turned by three x to the fifth, the denominator would be one. So there will be no more denominator one minus five over three. Right. This differently. Fire over three X squared prince to over three excessive force. It means that the M same they functions, have two seamen behavior has the party and we were able to also show this graphically in party.

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