Problem 61 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^4 - x^6 $

Answer

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Video Transcript

this brown number sixty one of the Stuart calculus. See, if Edition section two point six and the limits has X approaches Infinity and his X approaches Negative. Infinity uses information together with intercepts to give you a rough sketch of the graph. The function is why equals excellent forthe minus X to the six. Um, we're going to take our limit first, that sex approaches infinity and we need to reduce form of the function we're gonna factor out and makes of the fourth term. And that leaves us with one minus X squared. On this term, we can simplify even further as one minus X squared. Is this quick because the difference of squares and that leaves us with from one minus X and one plus six. Okay, as experts in Infinity, we know that this limit their properties, elements allows us to take the limit individually each term. It's well correct. And do that on the limited express infinity of ex fourth. It's going to be Kennedy multiplied by one minus a very large number. It's going to be negative infinity and the last term is one plus a very large number infinity. So out of three very large numbers being multiplied together yet another large number. And I think it is time to positives gives us a negative. So this limited purchase negative in committee as experts Negative infinity of the same function. Thanks to the forest One of my ethics times one plus x Here our limit If we do it again individually because their properties limits allow us to do that. Negative infinity. A very large negative number to the fourth Power is a very large positive number one minus a very large negative number is a very large positive number and then one plus a very large negative number is a very large negative number. So this limit a purchase negative to me as experts in negative infinity. Our next steps are to find the X Y intercepts the Y interceptors where X is equal to zero. And if we plug in our ah value zero for X, we get a zero to the forth my zero to the sixth for zero. So our why intercepted that the origin zero zero. And if we look for the X values that make y equals zero and we use our readies for him, exit the fourth one minus X times one plus six What is this? Equal to zero? The answers are the ex intercepts are zero positive one and a good one. So three ex intercepts one Why intercept? And we can go in and sketch what this craft may look like Point Our points of interests are native one the road No one. We know that as the function approaches negative infinity and positive infinity The functional virtual native infinity. So have this shape, you know above The XX is here. Touchdown! Here! The origin! Go back up and then come back! Yeah! So our function is of this form. We see that it clearly it's one way Intercept three ex intercepts and zero one and everyone and that the limit has X approaches. Infinity and negative Infinity both approached. Negative! Infinity on That is our final answer.