Chapter1FunctionsLimitsand$ 1.7 Two Remarkable Limits
Chapter 1 Functions and Limits §1.7 Two Remarkable Limits
IntroductionI.SqueezeTheoremsinxlimI.x0Xm.Monotonic SequenceTheoremIVlim(1+=)*=ex-80十S 1.7 Two Remarkable Limits
§1.7 Two Remarkable Limits I. Squeeze Theorem Introduction III. Monotonic Sequence Theorem ) e 1 lim(1+ = → x x x IV 1 sin lim 0 = → x x x II
I.SqueezeTheorem1.SequenceSuppose thatt lim y, = A,lim z, = A,n80, ≤ x, ≤ zn,for n ≥ K (K is a fixed integer)Then lim x, = A.n-→0S 1.7 Two Remarkable Limits
§1.7 Two Remarkable Limits I. Squeeze Theorem 1.Sequence y x z for n K (K is a fixed integer). y A z A n n n n n n n = = → → , Suppose that lim ,lim , Then x A. n n = → lim
I. Squeeze Theorem2. FunctionSuppose thatlim g(x) = A, lim h(x) = A,x-→ax→ax-→>00X-→01g(x)≤ f(x)≤h(x),as xeU(a,)(x|> M)yThen lim f(x) = A.x-→ax-→00h2gx0aS 1.7 Two Remarkable Limits
§1.7 Two Remarkable Limits 2. Function lim g(x) A,lim h(x) A, x x a x x a = = → → → → Suppose that Then f x A. x x a = → → lim ( ) g(x) f (x) h(x),as x U(a, )( x M). 0 o x y a A g f h I. Squeeze Theorem
Example12"Provethat=0limn!n-→8Proof2"222242.2...20<L132n!1.2...nnn42"since= 0,lim 0 = limthereforelim= 0.n→00n-→o nn!n-→>2″4Wrong!lim0<limlimn!n-00n-→0n-0nS1.7 Two Remarkable Limits
§1.7 Two Remarkable Limits Example 1 0 ! 2 lim = → n n n Prove that n n n = 1 2 2 2 2 ! 2 n 2 3 2 2 2 1 2 = n 4 0 Proof 0, 4 lim 0 = lim = n→ n→ n since 0. ! 2 lim = → n n n therefore n n n n n n 4 lim ! 2 lim 0 lim → → → Wrong!