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Problem 80 Medium Difficulty

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

Answer

$\lim _{x \rightarrow-\infty}\left(1+x^{3}\right)=-\infty$

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Video Transcript

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.