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Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{(2x^2 + 1)^2}{(x - 1)^2(x^2 + x)} $

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we want to find the limit of this expression as X approaches positive infinity. Now, when X gets very large on its way towards positive infinity, uh the numerator will be dominated by a particular term and so will the denominator. Um In the numerator we have two X squared plus one being raised to the second. When we look at the expression two x square plus one, the dominating term is going to be two X squared. And that's because as X gets very large towards positive infinity uh two X squared plus one will be dominated by two X squared effects is very large. X squared will be very large times in it by two will be very large adding one to that will not increase it that much. So the dominating term uh in this polynomial is two x squared. Uh So the top part of this function, the top part of this function is going to be dominated by the two x squared term. And keep in mind that that has to be raised to the second power. So we'll look at that in a moment. Yeah. Now in the denominator we have x minus one squared times X squared plus x in the x minus one. Uh expression, the dominating term clearly is the X as X gets very large X is going to get very large subtracting one from, it's not really going to change it. So the X -1 expression is dominated by the x term and then keep in mind we will have to square it. Now in this uh expression expert plus X. Uh the dominating term is going to be X squared because as X gets very large X squared gets even larger because she times in it by itself adding it to X does increase it because X is getting large but X squared is a whole lot more larger bigger than X because you're taking a large number and your times and by itself. So the dominating a term in this expression is two X squared. Uh So the numerator is being dominated by two X squared and there has to be raised to the second. Uh The denominator is being dominated in this expression by X, which has to be raised to the second and that's time zing. Uh The dominating term in this expression which is X squared. So to make a long story short, the numerator, this entire expression two X squared plus one to the second will be dominated by two X square to the second. The numerator is going to behave largely as the two X square to the second behaves now, two X squared to the second is really to to the second which is four times X squared to the second which is X to the fourth. So if X is moving towards positive infinity, the numerator is going to be behaving like four Times X to the 4th. The denominator is going to be behaving as X squared times X squared does X squared times X squared is exited 1/4. So the denominator of this entire expression will behave as X to the fourth does uh as X approaches positive infinity now, four times X to the fourth, divided by X to the fourth. Those exit a force will cancel And that leaves you with four. And so the limit of this entire expression as X approaches positive infinity is equal to four. This entire expression will approach the value of four as X approaches positive infinity.