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 = 2x^3 - x^4 $
This is problem number sixty of this tour. Calculus eighth edition, Section two point six. Find the limits. His ex approaches Infinity and his expertise negative. Infinity uses information together with intercepts to give a rough sketch of the graph as an example. Twelve. The function is why equals two. X cubed minus X to the fourth. Our first step will be to find the limit as experts is infinity, and we're going to take this function to excuse my fourth gonna simplify it a little bit. Factor out in X cubed term That leaves us with executed multiplied by the quantity to minus X. And if we consider X has a very large positive number offers, we should clarify through properties of limits that the limit of a product is the same as a product of limits. So it will be the limit of X cubed as express infinity multiplied by the limit kind of to money, sex and six purchase infinity. And at this point, we can ah see clearly. The first limit is infinity beverages towards Infinity Cube is an increasing about function so that we can approximate Is it going towards infinity as X close approaches infinity The second limit. We have two constant minus and number that's getting larger and larger. So if we pick a very large numbers, say on one million, two minus one million is almost negative. One million and his purchase infinity. This will become increasingly negative and larger. So we say that this is a limited purchase of negative infinity. And if we have a very large positive number multiplied by a very large negative number well, this is just still going to be a very large emerges. Negative has deposited him too negative equals negative. So this is the behavior of the function as expert sis impunity, the function of purchase Negative infinity. If we are to repeat the same steps for ex protein negative infinity, we'd be able to reduce the function as we did previously, and separate the limit as we did with our property limits. But this time we're doing a limited sex approaches. Negative infinity! This time the second limit is the one that I purchased. Infinity since two minus a negative number is positive numbers, so it gets increasingly large and then a negative number cute is still negative. So this let me will approach negative infinity. And we actually have the same result for this limit that the function of purchase Negative Infinity. When the When at when The X value purchase Negative infinity for our intercepts D y intercept occurs. Ah, when it wherever Whenever X is equal to zero. So where next is equal to zero? That's right. This in a different color the next zero Our function is two times zero cubed minus zero to the fourth or a zero You are why intercept is equal to zero. Our X intercept will be wherever y zero and one step we could do is use the fact of form execute a more quiet right two minute sex. And here we identified that y zero whenever x zero or when Exes too. So why intercept? His zero x intercepts are zero and two. We can strong approximately sketch approximately a dysfunctional looked like we have some Pacific point zero and it too This function specifically, uh, we'LL have in this shape and here on this last part let's try it again. Here we can see that the Lim ah values that we got our consistent as X approaches infinity this being ex this being wine as X approaches Infinity heart function Purchase Negative. Infinity as expert is negative. Infinity, Our function to purchase negative infinity And our one y intercept is here at y equals zero. And our two ex intercepts are here and here at X equals zero and addicts equals two. Ah, and this is what the function looks like as a rescue JJ.