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(a) Use a graph of$$f(x)=\left(1-\frac{2}{x}\right)^{x}$$to estimate the value of $\lim _{x \rightarrow \infty} f(x)$ correct to two decimal places.(b) Use a table of values of $f(x)$ to estimate the limit to four decimal places.

A. $$\lim _{x \rightarrow \infty}\left(1-\frac{2}{x}\right)^{x} \approx 0.135$$B. $\lim _{x \rightarrow \infty} f(x) \approx 0.135$

Calculus 1 / AB

Chapter 1

FUNCTIONS AND LIMITS

Section 6

Limits Involving Infinity

Functions

Limits

Continuous Functions

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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Susie function is just part of a function. Uh, FX is equal. Teoh, Uh, one Y minus two over X to the Power X. This is just part of it, right? There's another part like that. But I'm not interested in this one because I'm looking at the limit as X approaches infinity. And this is the part that we're looking at because extrapolation infinity is like this part, right? So that is why I did not concern myself, but he had a part like that, right? So I'm just concerned myself with a right side because of those love it right as X approaches infinity of this function. Right? So what is the limit as X approaches of remedial the function, um, one minus two over X to the power X year, right? What is what is that limit? So when you zoom in your calculator, you can use your graphing calculator or you could use on online graphing calculator right under Internet. When you zoom any and you can see that here, you trace it to the white value. Consider here is approximately what, 0.135 right? Because it is at this point that the current levels out. Curious level in is almost horizontal here. Right. So if you zoom in your calculator, what do you use an online calculator or using a graphing calculator yourself? You can see that it's approximately 11 and out at 0.135 Right, Because this is super in one, right? And 0.15 So 0.135 is somewhere in between, right? So when assuming probably can see that this momentous approximately 0.135 Right. And we can also do finding by the table for right every face you can plug in values of X into this function to see where it is going. Right. So when you plug it into a table for what you can start from 10 you know 1000 you know, 100,000 and all that can go on and on because you're approaching infinity, Right? Richard Infinity here. So you could go on and on and on and on. And each time you do that, say, if you do 10. This is Sue 100.1, uh, 07 073 right. Uh, seven. You know, if you do 1000 right, it could continue in 2000 1000 is number 13500.1350 Right. And then if you get get to like, ah 100,000 you still get it. 0.1353 threes. He'd so you see, it is still in the neighborhood of 0.135 Right? So you can see that as you go on and on and on, you get in something that is in the neighborhood of 0.135 approximately. Right. So if we use a table, gives you that firefighters 120,000 1 million do continuously. You can see that you be getting in the neighborhood of 0.135 Right? So therefore, by the tables, you could also see that the limit The limit here is true, right? So the graph in a table uses the same arrives

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