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Determine the infinite limit.

$ \displaystyle \lim_{x \to 0}(\ln x^2 - x^{-2}) $

$\lim _{x \rightarrow 0}\left(\ln x^{2}-x^{-2}\right)=-\infty \sin c e \ln x^{2} \rightarrow-\infty$ and $x^{-2} \rightarrow \infty$ as $x \rightarrow 0$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Limits

Derivatives

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This is problem forty three of a Stuart Calculus eighth edition Section two point two determined the infinite living the limit as X approaches zero from the quantity Alan of X Word minus X to the negative, too. We'LL begin Henry about re writing the limit slightly differently. Clement and experts Zero huh? The natural log of X squared minus. We will change the negative exponents as a reciprocal one over X squared. Now we will treat each of these terms separately, prime the approximate limit of each and hope that the limit converges afterward. We'LL begin with the first term for this first term ex approaching zero. Notice that there is no specification, whether it's from the left or the right. Luckily for this problem, that distinction is unnecessary. Seen as X squared for each turn each time that X is included, it is squared, meaning that whether it is a negative number or a positive number, this number will always be positive. And so we will see that that won't affect our solution. So, for example, here, if X is approaching zero, it is a small number. It could be a small negative number. Corn. It could be a small positive member. Either way, when it is squared, it is a very small, positive number. And we know that this ratio of any finding number of anybody small, positive number give Mrs value of infinity as it tends towards infinity. Looking at this locker at the MC function, we know that just as we show just was as well as we showed it here on the right, taking a value very, very small, close to zero and scoring the value a matter of its hespe, all negative number or a very small positive number When it's squared, he still remained. What a very spawn, a positive number. And if we can recall, Okay, honey, graph of the natural longer function as X approaches zero. We know that the function of purchase negative infinity. And so we take that solution, you place it here and then we make sure to remember this thing out of his wound. And we have a negative infinity that were resulted from this natural algorithm as ex took tends toward zero. This natural algorithm tends toward saying infinity minus one directed by a very small number squared. This tends towards positive infinity a negative infinity minus B positive. Infinity is Avery large negative number. And therefore this limit tends towards negative infinity. If we recall a plot here of this function I Linda X squared minus thanks to the major too. Which was the original problem statement. We see that whether it's from the left or the right, disfunction tends towards the negative infinity. So we can defend Italy State that the limit exists because lim from the left I think the infinity the limit from the rightest father is negative infinity and therefore the limit of dysfunction is negative Infinity.

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