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Find the limit or show that it does not exist.
$ \displaystyle \lim_{x \to -\infty}\frac{4x^3 + 6x^2 - 2}{2x^3 - 4x + 5} $
2
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 6
Limits at Infinity: Horizontal Asymptotes
Limits
Derivatives
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this is problem number eighteen of Stuart Accuracy Addition. Section two point six Final Limit. Our show that it does not exist element and sex approaches. Negative Infinity Oh, for execute six X squared minus two about it, by the quantity to execute. Fine. It's for X Plus five and we The way that we are approaching this element is Bonnie simplifying the women itself. Is it using the approach of converting our limit into one of this format where the limit his ex porches? Negative affinity, for example, of the quantity one of the exhibit Our power, where are is a rational value greater than zero. Is he cool to zero? So in order to do that, we're going to take the largest X to a power that exists in the numerator or denominator, and divide each term by that turn. So, for example, for X for each term we're going divided by X cubed. The first term, for example, for X cubed minus divided by four X cubed is just for thie. Next term will be six x word about about X cubed, which just leaves six or X and then the final term in the numerator is going to be, too to buy the right execute and the denominator we're going to execute about. I execute just leaves too for X, divided by X cubed is just four or x squared. And if I only in the last term that in our mayor will be five or excuse and in order for us to be able to use this limit on a definition here which will allow us to eliminate some terms if they look like one of us, you are. We're going to use their properties and limits to separate this fraction into a limited the numerator divided by the limit of the nominator and then use the properties of limits of sums and differences to separate the limit into a sum and different elements. So we'LL have ah limited six person negative infinity of four plus limit is expert disability a six or X minus The limited express infinity of two of our x cubed all in the animator and the denominator will be the limit is exporting to infinity of two and as long it is X approaches infinity for over X squared. Tell us the limit is experts Infinity hour five over execute And since each of these limits are the same as the original limit sex person in an affinity and, ah, there are various terms that looked like this function here one of Rex to the O. R. We will be able to reason that measure these limits approach zero as experts negative affinity and so, for of these tears will cancel out on. And then we review the limit of expression, affinity for it thes constant value. Sworn to ltd re constantly is just constantly so then a medal before the denominator will be, too. And our final answer for this and it will be two.
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