Problem 62 Hard Difficulty

Find the limits as $ x \to \infty $ and as $ x \to -\infty $. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

$ y = x^3(x + 2)^2(x - 1) $

Answer

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this problem Number sixty two of the sewer Calculus eighth edition Section two point six Find the Limits has expertise in vanity and his experts. Negative Infinity is this information together with intercepts to give a rough sketch of the graph? Has an example twelve and the function is why equals X Cubed times the quantity X plus two squared times the quantity X minus one. So our first limit and six persons infinity will be exactly this function limited dysfunction on DH We have a product of a few terms executed plus the quantity X plus two squared plus the quantity experience one. Our properties of limits allowed us to do each of these limits individually. So the first term a purchase positive infinity as experts is positive Infinity this second term Also purchase positive infinity And then we didn't have to hear a very large number acts approach in a very large number minus a very small number one. Also purchase positive affinity Therefore, her own this limited purchase infinity When X approaches infinity what is the limit? Is experts negative could be about the same function Fine. We take the same function and we're going to play the properties limits that allow us to take that individual limits and multiply them together. Negative Infinity If X is a person negative infinity and that number is cute It'Ll remain negative and a real very large number Infinity The second term It's a very large negative number squared. That is a very large positive number. And finally, in a very large negative number minus one, it'LL still remain a very large negative number And so we have a very large negative number of times. Every large number positive number comes another very large negative number. So this also approaches positive infinity. So both the limit as they could purchase infinity and eliminate six Purchase negative infinity They both approach positive infinity for our X y intercept Are y intercepted were X equals zero. So go ahead and play this hand What do we get when X equals zero? We'LL get there a cute don't swear by their applause to squared Well, the part about zero minus one We get zero times two squared or just for times negative one which is overall zero to the the Y intercept Is it the origin Where is like one zero? That's for finding ex intercepts. We go ahead and set of this problem. Execute him was a quantity X plus two squared times X minus one and then asking Where is this? Equals zero? Well, this is equal to zero. Whenever each of these terms individual air equals zero that occurs at X equals zero and X equals negative too. And then X equals positive One toe, three ex intercepts. Well, go ahead and provide a very rough sketch for this craft. We just said the extender ships are at Ah, negative too at zero. And that one. We know that the function of purchase infinity as explosions, *** infinity and positive infinity. So I'll have this shape two left into the right of the eccentricities. One in negative too. Ah, tea. The function will proceed up. We're this way between negative two and zero slightly. Then come back down towards zero. This is where the other X intercept ins come back down a little bit and then go back up through Thanks. Intercept of one two. This is where the function should look like. The extender ships are clearly shown at one zero. Negative too. The way I intercepted that y equals zero as X approaches positive from it either function of purchase positive infinity and his expression that infinity the function also purchase positive Infinity on DH that is all consistent for this craft.