121616949-math.49 - 2.4 The Derivative Function 35 1(Hint...

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2.4 The Derivative Function 35 16. lim x 0 x sin 1 x « (Hint: Use the fact that | sin a | < 1 for any real number a . You should probably use the definition of a limit here.) 17. Give an δ proof, similar to example 2.4 , of the fact that lim x 4 (2 x - 5) = 3. 18. Evaluate the expressions by reference to this graph: (a) lim x 4 f ( x ) (b) lim x →- 3 f ( x ) (c) lim x 0 f ( x ) (d) lim x 0 - f ( x ) (e) lim x 0 + f ( x ) (f) f ( - 2) (g) lim x 2 - f ( x ) (h) lim x →- 2 - f ( x ) (i) lim x 0 f ( x + 1) (j) f (0) (k) lim x 1 - f ( x - 4) (l) lim x 0 + f ( x - 2) 19. Use a calculator to estimate lim x 0 sin x x . 20. Use a calculator to estimate lim x 0 tan(3 x ) tan(5 x ) . We have seen how to create, or derive, a new function
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Unformatted text preview: f ± ( x ) from a function f ( x ), and that this new function carries important information. In one example we saw that f ± ( x ) tells us how steep the graph of f ( x ) is; in another we saw that f ± ( x ) tells us the speed of an object if f ( x ) tells us the position of the object at time x . As we said earlier, this same mathematical idea is useful whenever f ( x ) represents some changing quantity and we...
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