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(a) How large do we have to take $ x $ so that $ 1/x^2 < 0.0001 $?

(b) Take $ r = 2 $ in Theorem 5, we have the statement $$ \lim_{x \to \infty} \frac{1}{x^2} = 0 $$Prove this directly using Definition 7.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

University of Michigan - Ann Arbor

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 problem Number seventy five of this tour. Calculus. Eighth edition, Section two point six, part eight. How large do we have to take X so that one over X squared is less than zero point zero zero zero one for this part? We're going ahead and going to solve this outbreak. Lee s. So when is this true? One over X squared. Less than point zero zero there. One that's the same Assane one over X squared. Ah, less than ten thousand or one over ten thousand. Now our next step will be to take the reciprocal of Kotite's changing this the direction of the inequality. Ah, and finally we take a square. So in this case, we need to take X as large as one hundred. In order for us to confirm that one over X squared is less than a one one one ten thousand X must be larger than one hundred. Prepare B. Yeah, according to deer and five we take, our is equal to two. So that means the exponent on the X and the denominator is too. We have this statement the limit as ex purchase infinity of the function one over X squared is equal to zero. Prove this circle using definition seven definition seven says if X is greater than the value and then the distance or the difference between the function on the limit will be less than a certain no margin. A certain error Absalon came. So we need to be able to confirm this is true for any choice of Absalon that we can find a certain end value s so we can proceed and use what function we have here. One over x squared, the limit l which is zero in this case less than Absalon. We're trying to figure out maybe what absolute should be. We're going to restrict Absalon to only positive values, which means that can disregard the absolutely sign. Ah, we can take the reciprocal X squared greater than Absalon and ah, or ex ex Greg greater than one over. Absolutely. And then finally the square root ex greater than one over the square root of Absalom. At this point, this is what satisfies this second part. The absolute value, the quantity effects, F X minus l being less in a quantity. Absalon. We also see that this has to be true for X is greater than a certain number. And so here we conclude that if we choose and to be at least one over the square root of Absalon Ah, then that confirms the statement. So for any Absalon, we can choose ups on to be one and choose Epsilon to be point one point zero zero one every single time. Well, we will be able to choose an appropriate and value such that this statement is true and according to definition seven, because that is true. This limit is in fact, true that the limited his expressions affinity of one over X squared is equal to zero.

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