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Formulate a precise definition of $$ \lim_{x \to -\infty} f(x) = -\infty $$ Then use your definition to prove that $$ \lim_{x \to -\infty} (1 + x^3) = -\infty $$

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Missouri State University

Campbell University

University of Nottingham

Boston College

Lectures

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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this barrel. Number eighty of this tour calculus eighth edition, section two point six Formulate a precise definition are the limit is X approaches negative Infinity. The function of X is equal to negative immunity. In our definition, I will hold for values and greater lesson zero and listen. Zero. So for every m there existed and such that if xs Liston's value m n the function is guaranteed to be equal to negative infinity. So ah, we're going to start with this second part here. The function one plus execute less than about em and then proceeded to solve for X less than M minus one. And then, at this point, we could take the Q Bert of about minus one, confirming that this is a negative value since Emma. Negative, very large negative value are any negative. Allie minus one is still negative. Curative. A negative values negative. So all that is considered. This is a singular value Q brew. And this corresponds to this other condition except lesson in so we can choose and to be Cuba tive and minus one. And this guarantees that for every M, we can find it in such that the conditions are met for our precise definition, and this is exactly the way that we have to prove it. This is our definition that we stated we have. This is an example and equals to the cube root of this quantity, minus and minus one that guarantees that as a limited, a CZ expertise, negative infinity of thiss function one plus x cubed. It's definitely equal to native affinity. This's true, based on our definition, that we have shown this proof.

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