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Rank the functions in order of how quickly they grow as $x \rightarrow \infty$$$y=x^{5}, \quad y=\ln \left(x^{10}\right), \quad y=e^{2 x}, \quad y=e^{3 x}$$

$y=e^{3 x}, y=e^{2 x}, y=x^{5}, y=\ln \left(x^{10}\right)$

Calculus 1 / AB

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

Differentiation

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Okay, So here were given these four conscience, Why is equal X to power five? Why is equal to the natural log of X to the power of 10? Why is equal to e to the power of two times X and why is equal to eat? The power of three X is we're going to use this property that for two functions f of X and G of X. If we consider the ratio the limit as X approaches infinity of F of X over G of X. If this is equal to zero, then that means F grows more slowly, then G. And if this is equal to infinity than F grows faster than G. Okay, so we'll just have to make this comparison evaluate this limit over and over and over again with these different pairs to find out the order. So we will start with ah to that seem kind of easy to me. Uh, we're gonna take the limit, Has X approaches infinity of E to the power of three X over E to the power of to Excite. Chose these because we get a really nice cancellation, right? We're gonna get the limit as X approaches infinity. So either the two X cancels out of simply e to the X. And so when we evaluate this limit, we're gonna get you to the power and fanny, which is equal to infinity. So that tells us the e the power of three X is faster than eat the power of two X. Okay, so we'll hang on to that fact and move on to our next comparison s. I want to compare eat part of two X to something else then and so I'll take the limit as X approaches infinity of e to the power of two X over X to the power of five. So if I evaluate this limit, you know, we get E two part of two times infinity over infinity to the power of five. This is infinity over infinity, so we can go ahead and use Blue Beetle's rule. And so this is equivalent to the limit as X approaches infinity of two times e to the power of two x over five x to the power of four. But again, I'll get the same thing. When I evaluate, I get infinity over infinity. Somebody used low petals rule again and get the limit as extra approaches Infinity of four times E to the power to x over 20 x cubed And I can simplify this so we can get to the fore and the 20 will become for over 20 is gonna be the same as 1/5. But evaluating this again and we're gonna get infinity over infinity. Ah, so we are going to rewrite it again and this is the same going to give us the same thing as the limit. As Ex approaching infinity of two times e to the power of two X over 15 x squared again evaluated This is gonna give us infinity over infinities So we will pilo Beatles rule again Take the limit as X approaches infinity of four times E to the power of two x over 30 x We evaluate this we're gonna get infinity over infinity again so we'll probably tells rule again to the limit As X approaches, Infinity will get eight times e to the power of two acts over 30 And this time when we have pilot Beatles rule we are going to get infinity over 30 which is equal to just infinity right. So this tells us that e to the power of two X is faster than X to the power of five. Okay, so from that we know we have excellent power of five is going to be slower than e to the power of two ex just going to be slower than e to the power of three X. So we don't try to compare our last one thio X to the power of five. If it is slower than we know, it goes here. But if it's faster than we were gonna need to compare it, uh, to each of the two x Andy to the three X as well, let's go ahead and take the limit as X approaches infinity Ah x to the power of five over the natural log of X to the power of 10. When we evaluate this, we get infinity to the power of fire which is infinity, and the natural lot of infinity, which is infinity. So we use low patrols rule. This will be equivalent to the limit as X approaches infinity of five x to the power of four over and then we'll have 10 x to the power of nine times one over X to the power of 10. Just going to simplify to the limit as X approaches infinity of five X to the power of four over. And so most these exes cancel out. There's gonna have 10 over X and this can simplify even further. This is going to be the same as the limit as X approaches infinity of who have 5/10 which is 1/2. But then this X flips up to the numerator. So we really have extra power five over too. Okay, When we evaluate this limit, we simply get infinity, right infinity over two, which is the same as infinity. So that tells us that X two out of five is faster than the natural log of X to the power of 10. So that means if we go back to this one since X to the power of five is faster, then the natural log of X to the power of 10 would be the slowest. And then this is going to be the order order of speed that we get with E to the power of three X being the fastest and the natural log being the slowest. As for slow and fast

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