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lecture4_1

# lecture4_1 - 2.4 One-sided Limits and Limits at Infinity...

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Unformatted text preview: 2.4 One-sided Limits and Limits at Infinity One-Sided Limits One-sided limits are limits as x approaches the number x0 from the left-hand side (where x < x0 ) or the right-hand side ( x > x0 ) only. Example 1 The domain of f (x) = 4 - x2 is [-2, 2]; its graph is a semicircle. What is limx-2+ 4 - x2 and limx-2- 4 - x2 ? THEOREM 6 A function f (x) has a limit as x approaches c if and only if it has left-hand and right-hand limits there and these one-sided limits are equal: 1 Example 2 limx0+ f (x) = limx0- f (x) = limx0 f (x) = limx1- f (x) = limx1+ f (x) = limx1 f (x) = limx2- f (x) = limx2+ f (x) = limx2 f (x) = limx3- f (x) = limx4- f (x) = limx4+ f (x) = 2 Precise Definitions of One-Sided Limits DEFINITIONS We say that f (x) has a right-hand limit L at x0 , and write if for every number that for all x > 0 there exists a corresponding number > 0 such We say that f (x) has a left-hand limit L at x0 , and write if for every number that for all x > 0 there exists a corresponding number > 0 such 3 Example 3 Prove that limx0+ x = 0. Example 4 1 y = sin( x ) has no limit as x approaches zero from either side. 4 Limits involving THEOREM 7 sin x x sin x = x (x in radians) x0 lim Proof From geometry (see book page 88) we get 1 1 1 sin < < tan . 2 2 2 1 < sin cos sin 1> > cos . 1< Sandwich Theorem: 0 lim - sin = 5 Example 5 Show that (a) and (b) cos h - 1 =0 h0 h lim sin 2x 2 = x0 5x 5 lim 6 ...
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lecture4_1 - 2.4 One-sided Limits and Limits at Infinity...

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