Problem 63 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 = (3 - x)(1 + x)^2(1 - x)^4 $

Answer

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Video Transcript

this problem. Number sixty three. The sewer Calculus eighth edition Section two point six Find the limits. His ex approaches infinity and his expertise is negative. Infinity These information together with intercepts to give a broth sketch of the graph the function is has given why equals the quantity. Three. Money sex turns a quantity. Want those x squared times a quantity one minus X to the fourth power. Our first step is to find this limit and Zac's approaches in committing, and we're going to take advantage of the property of limits that allows us to take a limit of each individual term, since each of these terms are being more complied. Ah, the answer will be the answer to this limit of product of a product of three terms will be the product of three separate limits. The first term three minutes X approaches negative. Infinity is expert is infinity The second term purchase positive Infinity as experts is infinity and the last term approaches positive Infinity also Ah, as except riches infinity. Since this will be a negative number, however, to fourth power or mean positive. So overall we have a negative infinity times to bother affinities and that comes out approximately towards negative infinity. So this whole limit with that function as experts is infinity will be positive Infinity as experts negative infinity of this same function all again take advantage of the properties of limits and apply a limit to each term individually and then want to play those at the end. So the first term approach is three minus negative Infinity So that a purchase Positive Infinity Very large Positive number Second term approaches one minus one Pleasant native, Very large number This is follower Negative, but it's square too so that we'll remain positive and this last term one minus a negative number is always positive Ah, there is positive in this case because our number is very large and this very large positive number to the fourth power is still a very large positive number. So three positive numbers multiply out a large positive number All these three infinities Ah come out to be a positive Anthony And so the limit is experts is negative. Infinity is positive Infinity for this election. Now to find the exit liner ships the Y intercept the Y interceptors where X equals zero. So we're going to go ahead and play that in. What is the functional I equal to when we put in zero for eggs and his hero look. Zero squared one minus zero before that gives us three times one squares. That's three minutes times one to the fourth word That's just three. Um, the weiners are the Texas interceptors were y equals zero. So we go in and just set this function equal to zero. And for this function to be zero each of these terms, at least one of these sinners must be zero. So there's three different ways we can do that. X equals three. Makes the first term zero X equals negative one makes the second term zero and X equals positive. One makes that their term zero. So those are your three ex intercepts on finally using all this information putting all this together and come up with a direct route sketch For with this function to look like we have prints of interests that negative one no. One and three the function as expressions negative Infinity is positive infinity So it'LL go on destruction The function as experts is positive Infinity is negative infinity so from three honor should pursed this direction. Ah here When the function of purchase native won, the function will turn back up And for each hour Why Intercept of Wyc owes three We'll do that Come back down towards the other X intercept that one So this is the wine is up the tree and then after this point goes back up again and then it comes back down there. So roughly speaking, the function will look like this. Notice that it's consistent with all of the ex intercepts and at NATO one one and three the y intercept at three and the two limits Expert's infinity The function parts negative Infinity as expert is making infinity their function Purchase infinity.