This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 2.4 Onesided Limits and Limits at Infinity
OneSided Limits
Onesided limits are limits as x approaches the number x0 from the lefthand side (where x < x0 ) or the righthand side ( x > x0 ) only. Example 1 The domain of f (x) = 4  x2 is [2, 2]; its graph is a semicircle. What is limx2+ 4  x2 and limx2 4  x2 ? THEOREM 6 A function f (x) has a limit as x approaches c if and only if it has lefthand and righthand limits there and these onesided 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 OneSided Limits
DEFINITIONS We say that f (x) has a righthand 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 lefthand 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
 Spring '08
 FBHinkelmann
 Calculus, Limits, Limit, lim, Limit of a function

Click to edit the document details