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Problem 16 Easy Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{1 - x^2}{x^3 - x + 1} $

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Video Transcript

So here we are given information about specific limit. We have the limit as X approaches infinity of one minus X squared over X cubed minus x plus one. Yeah. So mainly here, once again, since we're evaluating X as X approaches large numbers, the constant terms are relatively going to be insignificant. And also when we compare X cubed and X X cubed is going to be significantly larger than X. So we can ignore X as well. So in the end this would just be evaluating the limit of the form negative X squared divided by X cubed and we can cancel out X squared from the numerator and denominator. And we just end up with negative one divided by X. So this is equivalent to negative one divided by infinity. So one over the within these just zero. So this would be our answer. However, we can also evaluate this by lawful rule which involves differentiating the numerator and denominator. However, this would just require more iteration. So it'll be -2 X. This will be three X squared minus one. Then we would have to do another iteration of law of the rule where we can find we would have negative too and this will just be six X. And once again we can apply Direct substitution and this would be negative to over six times infinity Which would just be equivalent to zero and this is our final answer