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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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03:05

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Find the limit or show tha…

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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

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