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If $ 2x \le g(x) \le x^4 - x^2 +2 $ for all $ x $, evaluate $ \displaystyle \lim_{x \to 1}g(x) $.

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Campbell University

University of Michigan - Ann Arbor

Boston College

suppose G of X is greater than or equal to two. X but less than or equal to x rays. Of the fourth power minus X squared plus two. Now, in this probably want to find the limit of the function G as X approaches one to do that, we first find me limit of the function two, X and X to the fourth power minus expert plus two as X approaches one. And so we have limits as X approaches one of two. X. This is equal to two times one, which is to and the limit as X approaches one of x rays to the fourth power minus X squared plus two. This is equal to one. Race to the fourth power minus one squared plus two. Or that's just too, which tells us that the limit as X approaches one of two X. This is equal to the limit as X approaches one of x rays to the fourth power minus X ray plus two. And this is equal to one. Know that in squeeze theorem, if we have F of X less than or equal to G of X less than or equal to H of X. And we found that the limit as X approaches A of F of X equals the limit as X approaches A of H of X, then the limit of G as X approaches mm will be equal to the limit of the two functions. Let's say this is equal to L. So this will also be equal to L. Therefore, by squeeze theorem, do you limit all the function G of X as X approaches one is equal to one?