121616949-math.44

# 121616949-math.44 - 30 Chapter 2 Instantaneous Rate Of...

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30 Chapter 2 Instantaneous Rate Of Change: The Derivative true whenever 0 < | x - 2 | < δ . So the question becomes: can we choose a value for δ that guarantees that 0 < | x - 2 | < δ implies | x - 2 | < ? Of course: no matter what is, δ = works. So it turns out to be very easy to prove something “obvious,” which is nice. It doesn’t take long before things get trickier, however. EXAMPLE 2.5 It seems clear that lim x 2 x 2 = 4. Let’s try to prove it. We will want to be able to show that | x 2 - 4 | < whenever 0 < | x - 2 | < δ , by choosing δ carefully. Is there any connection between | x - 2 | and | x 2 - 4 | ? Yes, and it’s not hard to spot, but it is not so simple as the previous example. We can write | x 2 - 4 | = | ( x +2)( x - 2) | . Now when | x - 2 | is small, part of | ( x +2)( x - 2) | is small, namely ( x - 2). What about ( x +2)? If x is close to 2, ( x + 2) certainly can’t be too big, but we need to somehow be precise about it. Let’s recall the “game” version of what is going on here. You get to pick an and I have to pick a δ that makes things work out. Presumably it is the really tiny values of
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