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speed-and-tails - You can ignore other terms 4 If a bounded...

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Growth Speed and Tail Thickness Math 175 - Fall 2009 Some Basic Functions. Ordered by speed of growth from slowest to fastest y = ln(ln x ) y = ln x . . . y = x 1 / 4 y = x 1 / 3 y = x 1 / 2 y = x y = x 3 / 2 y = x 2 y = x 3 . . . y = 2 x y = e x y = 10 x . . . y = Γ( x ) (factorial) y = x x Some Asymptotic Functions. Ordered by tail thickness from thickest to thinnest. y = 1 / ln(ln x ) y = 1 / ln x . . . y = 1 /x 1 / 4 y = 1 /x 1 / 3 y = 1 /x 1 / 2 y = 1 /x y = 1 /x 3 / 2 y = 1 /x 2 y = 1 /x 3 . . . y = 1 / 2 x y = 1 /e x y = 1 / 10 x . . . y = 1 / Γ( x ) y = 1 /x x Notes about other functions: 1. The vertical dots indicate other powers of x and other bases of exponential functions. There are many more complicated functions that you might encounter. You need to know where to put them on this list. 2. Constant multiples are irrelevant. 3. If two or more functions are added or subtracted the fastest growing term dominates.
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Unformatted text preview: You can ignore other terms. 4. If a bounded function appears as either a stand-alone term or a multiplier, it can be ignored. Some examples of bounded functions are sin x , cos x and arctan x and constant functions. 5. Products will live somewhere in between the functions on this list. For example, x ln x grows faster than x but slower than any higher power of x . 6. Quotients must be studied individually. More about this in class notes and later sections. 1...
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