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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Page1 / 6

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online