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Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{t \to -2}\frac{t^4 - 2}{2t^2 - 3t + 2} $

$\frac{7}{8}$

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in this problem we are asked to evaluate this limit limit T approaches -2 of T. The 4th -2 over to t squared -3 T Plus two. All right. So first of all, let's just check if we evaluate the denominator With T at -2. That's two times minus two squared minus three times minus two plus two. That is what we'll minus two squared is four times two is eight. Man is a modest. That's plus two times 3 is six plus two. So that's eight and eight. That's 16. Okay, so this is not zero. All right. So we don't have an issue with dividing by zero and having a vertical assam. Tote at minus two. So therefore the first limit law is at the limit of F of X. This little needle here half of X or G fx is limit of Absolutely limit of half of x times one over the limit of G f x. So that means that we now have a limit As T approaches -2 of Do you? The 4th 1 is two over The limit as T approaches -2 of to t squared minus three T plus two. Okay. Yeah, The next one is a polynomial limit law here. Right? Or the limit of a X squared plus bx. We'll see is is what is the constant? They can come out in front limit X squared plus B times the limit of X. Since he's a constant can come out front was a limit of C which is just see by the way. Okay, so that means I now have The limit as T approaches -2 of T to the 4th minus two over. The limit is two times the limit as T approaches minus two of t squared -3 times the limit as t approaches -2 of T plus two on the end. Okay, so evaluating each one of these limits, well, the limit of this polynomial is just Just going to be this number. Interview that ponytails, that's -2 to the fourth, homelessness to over two times A -2. Is that polonium? It's -2 squared -3 times -2. That's too. And so what do I get? Well minus to the fourth, that's 16 minus two, That's 14 over. We just calculated this to be 16 And so this is 7/8, isn't it? In reduced form.

Oklahoma State University