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Make a rough sketch of the curve $ y = x^n $ ($ n $ an integer) for the following five cases:

(i) $ n = 0 $

(ii) $ n > 0 $, $ n $ odd

(iii) $ n > 0 $, $ n $ even

(iv) $ n < 0 $, $ n $ odd

(v) $ n < 0 $, $ n $ even

Then use these sketches to find the following limits.

(a) $ \displaystyle \lim_{x \to 0^+} x^n $

(b) $ \displaystyle \lim_{x \to 0^-} x^n $

(c) $ \displaystyle \lim_{x \to \infty} x^n $

(d) $ \displaystyle \lim_{x \to -\infty} x^n $

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Oregon State University

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this problem. Number fifty six of the Sewer Calculus Expedition Section two point six. Make a rough sketch of the curve wine peepholes Exit Ian. If it is an insider for the following five cases, I N is equal to zero. Two I and it's greater than zero. And for an Izod three, Ari and his creator than zero brand is even four, and his lesson zero brand is odd and five and is listen zero for And it's even for the first case, we have the first part and equals it. Zero Queen of the function of one equals X to the zero power that is a constant value. And then a function that would look like that may look like a straight line just like this. A horizontal line and sort of be my equals. Tiresome. Constant. That's the first example. If it is odd. Ah, the function wood B symmetric about the origin. So something touch with this. Maybe one more time. Guys, what the function will look like for Part two and is greater than zero. And it's odd, and that's are rough sketch for that curve. If and his credit in zero and even it would be symmetric. But the wags is so maybe something like a problem. That would be a good example for Part three purports foreign fire we have in his great lesson. Zero. If it is odd, it may look like this where there is a clear asked him to and X equals zero, my friend is odd all the points in the positive demeanor and the positive positive domain plus plus and in the negative to me in their o negative domain for wine. Negative, Extinct. Why positive explosive line? And for the fifth graph, a friend is even No points are positive and there's still that ask himto lexical sirrah two. This is a rough sketch of all the five cases. And then we proceeded to answer each of these limits per case. Um, work on them one by one for the first curve. Why close to a constant as experts? Zero from the right, Um and actually, in this case, because in zero, it's not just a constant, it is exactly one for the same one specific case. So this would be like, I wonder, for example, that is a specific craft we're looking at. So the limited wanted purchase exports. Zero All right, Uh, next to the end is exactly equal to one. If you approach there from the left for exit the end to this crap, it is also equal to one. I've for part See if you're approaching infinity for this crap you are approaching one. There was constant. All the limits will be the same as experts. Negative unity. We'Ll also be one and that's for the first time for the second graph of second case If n is greater than zero and and it's hard as experts zero for this function from the right, the limit is equal to zero If x approaches from the left it is also equal to zero There you can tell go towards the origin for the limit is expert is infinity for part See, we see that the graph of purchase positive infinity and his expertise negative infinity for this odd function where n is greater than zero And approach is negative financing for this telegraph and is greater than zero And then it's even We will have the same results as a second case except for the last limit as we approach sir from the rate for party. Do you know how you function is equal to zero of the limit? Tell them it's expertise here on our left is also zero. If we approach affinity for party, the function her part three purchase positive, infinity and as experts is negative affinity for party. It also purchase positive Infinity For this third case for the fourth case, the Ltd's experts zero from the right for party. It is positive, infinity and sweet as we see we approach from the right. There's this vertical attitude and the function of perches. Very large positive number. Her part being eliminates experts zero from the left we see that for this fourth case where in these lessons here, when it is hard, the function approaches negative infinity for part, Seeing as express infinity, we see that the function gets closer and closer to the X axis but never reaches it. Which means that this function approaches zero and expertise infinity. And this is also the case for party. As the limited perches, Pretty limit has expertise in an affinity a function also purchase zero. Finally, the last case and his lesson zero for and it's even and we will have almost the same results except for the second limit. So First Party Limited's experts zero from the right that is part positive. Infinity Just like the previous case second limit for party limit its expertise here from the left is also positive. Infinity for this case as we see that both from the left and the right protein zero, it reaches positive Infinity Party Limited's expertise impunity. We see that the function of purchase the X axis therefore to purchase hero and this is also the case party. The limit is expressions negative. Infinity are the function Next the end also approaches zero And these air all the solutions to all these limits for these five cases presented into this problem.