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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^2(x^2 - 1)^2(x + 2) $

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Oregon State University

Harvey Mudd College

University of Nottingham

there's a problem. Sixty four Stuart Calculus eighth edition Section two point six Find the limits has expertise. Infinity and a sex purchase. Negative Unity uses information together with intercepts to give a rough sketch of the graph. The function is why equals X squared times the quantity X squared minus one squared times the quantity X plus two two. Our first question to answer what is element is expert disability. Ah, dysfunction. Why so he read out the function X square terms the quantity X squared minus one That quantity squared times The quantity was too and we'LL take advantage of our properties of limits. This is a limit of a product of terms, Andi. We can evaluate the limit individually. So our answer will be the product of the limit of each term tool to the limit individually and then multiply those together as experts in infinity X squared approaches positive infinity as experts impunity. As we said, X squared approaches Infinity minus one. It's still a very large positive number. Squared is still very large, positive number and this last term is linear. As expert infinity, this will also approach infinity. So the product of all these individual limits is infinity positive, which is a solution to that initial limit problem? We're going to do the same thing as expert is negative. Infinity still taking advantage of the property that limits that allow us to take the limit of each term individually and then multiply those results afterward the first term and x squared as experts Negative infinity. Ah, negative number squared is still is positive And so this number will approach positive infinity On the second term. As we said, this number will perch positive Infinity minus one. Still a very large positive number squared is still perch in positive infinity and then the last term here has expert is a negative infinity Again, this is a linear function. So as experts negative entity, this function will approach Take the infinity and overall, as you must provide two very large positive numbers and one very large negative number Yeah, how come it's a very large negative number So this limit second limit approach is negative Regarding the intercepts To find the Y intercept, we need to plug in X equals zero And if we go ahead and do that, we'LL get zero squared. Ah, you're a squared minus one quantity squared time zero plus two Since the first term in zero and it's being almost quite out. This one intercept is definitely at zero for the X intercept. They are the X values that make the function equal to zero. So why is it cold there? Here is the function As it stands, we're going to make one slight adjustment. Before we confirm all the ex intercepts, we're going to take thie middle term. Ah, and expanded out. Um, this will be the same as experiment is one of the difference of square. So it's at plus X plus one times X minus one In this whole quantity is square good. So this is important because we need identified that there are two explains that make this middle term equal to zero, both positive and negative one. And so we have four ex intercepts in this case, the first one the one the ex Valley that makes the first term zero zero the two terms and make the middle term equal to zero. Our positive one and negative one. And the last X Party that makes this term zero and therefore the whole thing. Zero is negative two. So there are four ex intercepts and only one wire it herself. So we're going to go ahead and provide very rough sketch of this graph Based on this information are points of interest are negative too. I need one positive wonder zero on those air for ex intercepts as well as their Y intercept Iria Y equals zero. And we know that his experts infinity on the first limit that the functional approach how that of infinity and six approaches Negative infinity This functional approach Negative infinity. So we'LL have this shape approximately on the function will increase a bit but then come back down towards the x axis that negative one afterward Chipper begin and then reached xx is that ah, X equals zero And then one more time we're going come back down And this is what the function looks like. Approximately one thing just to note in to use as a chick. Um, the reason that here we have the function touching X axis But not going through the X axis is either from positive to negative or negative to positive. Is that the specific ex intercepts thing to one zero one one? Um are turning points because those terms in this equation are raised to a even power. So, for example, Dex intercepts thing to one, and one of those air terms here that are raised to the second Power X squared is the term that provides the extender subject call zero that's raised the second power. However, on DH native to that term is raised to the first power, and therefore it does not. It is not considered a twenty point, so just a way to check the graph. Otherwise, dysfunction is definitely consistent with called the limits, and the extent accepts Hawaiian intercepts that we found on and therefore this is our final answer.