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(a) Prove that $ \displaystyle \lim_{x \to \infty} f(x) = \lim_{t \to 0^+} f(1/t) $ and $ \displaystyle \lim_{x \to -\infty} f(x) = \lim_{t \to 0^-} f(1/t) $ if these limits exist.

(b) Use part (a) and Exercise 65 to find $$ \lim_{x \to 0^+} x \sin \frac{1}{x} $$

a. This proves that $\lim _{t \rightarrow 0^{-}} f(1 / t)=L=\lim _{x \rightarrow-\infty} f(x)$

b. $$\begin{aligned}

\lim _{x \rightarrow 0^{+}} x \sin \frac{1}{x} &=\lim _{t \rightarrow 0^{+}} t \sin \frac{1}{t} \\

&=\lim _{y \rightarrow \infty} \frac{1}{y} \sin y \\

&=\lim _{x \rightarrow \infty} \frac{\sin x}{x} \\

&=0

\end{aligned}$$

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This is problem number eighty one of this tour calculus C edition Section two point six Party prove that the limit as expertise infinity of F of X is equal to the limit as T approaches zero from the right of the function evaluated one over tea and the Limited's expert isn't even Infinity of X is equal to the limit as T Bridges zero from the left of the function evaluated the one over team. If the's limits limits exist, so will begin with this first part here. It's a party. Let's say that this limit and six approaches affinity. However, FX. Let's call this limit now. Ah, now, from our definitions, this limit exists, Um such that for every Absalon created a hero there exists and great zero such that if X is greater than mm. The difference between the function and its limit is less than this value Absolute. Okay, so we have that has thie statement of the conditions of definitions for this limit that we stated that is equal to l. Okay, Our next part has to do with choosing tee. Let's let t equal. One of her ex came in a lot of this that follows is the based on this definition everything before, Um Then if thi's one rex then because X a greater than N or X is greater than n if and only if zero is less than one over X is less than one hour in. And this is just stating the relationship between X and N excreted then the reciprocal wanna Rex is less than one Iran and is restricted to a positive values. So this builder, these values air definitely greater than zero. But since we chose one of Rex to be t, this is equivalent to saying zero is less than tea is lesson one of our end and for convenience will call this condition one Iran Um, a new term. We're gonna give it to the signal took Sigma's ical toh wanna rent. So this is just a restriction from putting into context or t A S t is between some some restriction signem which is related to end by one or in and it's a positive value greater than zero. So through this definition ah, oui can prove in the same way. Or we can take this here and just modify it. The absolute value of as of actual ex is equivalent to one over team Through this tea is one of wrecks. Ex must be on our team. So one over t everyone on our team. I'm minus l. This will be true. Less than Absalon for for this condition. I should read it over here. Four zero's lesson Tease lesson Sigma. This is the equivalent statement to this previous definition here. So as long as we set all this up in this way, we can save any limit. It's just a a modified version of this limit here. Since we started out with and capital in half of X, everything was an ex. Now we want to state it in terms of tear. This limit for this new function half of one over team will be also equal to Ellen. Um And then instead of extra cools towards infinity. We see that as expressions of unity that t approaches zero. And it's important to also recall that tea is between zero and Value Sigma their forties and the positive domain s o as we approached you are approaching zero from the right. So overall we started with this definition. We came to it a new formulation of this limit in terms of team. And we've concluded that this function or this limit is equal to elm, which is exactly them original limit as experts infinity of FX and therefore we've proven on this first set of limits moving on to the second set, we're going to do essentially the same approach. So still per name, we're gonna kind of this limit as experts is negative. Infinity. No, no, on We have the same conditions that for an epsilon grid zero there exists and then in this case and will be less than zero because we're approaching a negative affinity. So it's that the difference between a function and the limited this absolutely this distance it's last time this absolute Valium if ah X is less than an hour. So this is the new condition for this limit here. So going off on that, we're going to take a simpler approach if he were going to let Teo on Rex on since ex lesson in we'LL accept is less than and if and only if not one, Aran is less than one rex. That's the relationship between Xidan. I'm just a reciprocal O'Ryan and since excess listen and and And this lesson zero these air to negative numbers. So they're both less than zero. And again we restate this. I'm calling one of her and a signal that's just a renaming of that term. One of Rex is team, and they brought less than zero. So he reached more or less the same steps. And we formulate the new definition for this you limit, Are we? We're formulating the definition for any limit, but we're using this past definition. So instead of FedEx ah, what tears want to Rex. Therefore, X is one over team so that one of the team the difference between this new function or this function and the limit is less than absolute. So we're just replacing ex with one of our team and what this line here shows. This is true for sigma Lesson t listen up. Listen zero This is a new formulation for a new limit different from the previous one because it's now a za function of tea rather than a function of X. It is the limit of the function one over team. Yeah, as T approaches zero from the left and we know this is true because we're having X approach negative infinity and his expression negative. Infinity t approaches zero power because of how we labelled everything Howe and his lesson zero and how Sigma One over and and tiara definitely less than zero. They were pushing zero from the left. And we have shown through this modified definition that this front limit is equal into Teo the original ml, meaning that it's exactly equivalent to the original image expert. You negative infinity of the function evidence. So we have proven those two sets of limits and they do indeed exist. And they're true in this form. We're going to use thes results for party now here in the middle. Ah, we're going to use these results from from party to solve this limit here in the middle, based on exercise sixty five to recall exercise, fifty five stated that the limit as exit purchase infinity of sign of X over X is equal to zero. So an exorcist sixty sixty five, we have proved this. So this is something we're going to take. It's given, and we're gonna take a look at this limit and see how we can use all of these tools to solve this limit. Well, this limit appears most to be like this first part here. This limit as t approaches zero from the right of a function one of routine. So I'm just going to rewrite it in that form, just gonna replace by T. This is just to make things look clear. T as T approaches here from the right of tea, Sinan won over eighteen way. We prove that this is equivalent to this. So we're going to change from a function of one of thirty tia proteins or from the right to a function of eggs. So instead of everything we see, one of the team will replace it with an X. Similarly, every time you see a tea will replace it with the one over X. And then our limit, instead of tea, goes to zero from the right exit approaching infinity. So we have changed our lim from how it was previously to how it is now. This is an equivalent statement because of what we proved in Party Limited's expertise. Infinity of sign of X rex is equivalent to the limit or China solved. And what we recalled from problem sixty five is at this limit is equal to zero. So, no, we have definitely confirmed through all of our proves in this problem, as well as the work and exercise to sixty five that the answer to this limit here of expert genes or from the right of X times sine of one rex is it zero?