Limits. Let y = f (t) be a function that gives the position at time t of an object moving along the y-axis. Then

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1 Limits From last time... Let y = f (t) be a function that gives the osition at time t of an object moving along the y-ais. Then Ave vel[t, t 2 ] = f (t 2) f (t ) t 2 t f (t + h) f (t) Velocity(t) =. h!0 h f(a+h) f(a) h a We need to be able to take its! a+h

2 Limit of a Function Definition We say that a function f aroaches the it L as aroaches a, written f () =L,!a if we can make f () as close to L as we want by taking su ciently close to a. f() L Δy Δ i.e. If you need y to be smaller, you only need to make smaller ( means change ) a One-sided its f(a) L r L l Right-handed it: L r =!a + f () if f () gets closer to L r as gets closer to a from the right a Left-handed it: L` =!a f () if f () gets closer to L` as gets closer to a from the left Theorem The it of f as! a eists if and only if both the right-hand and left-hand its eist and have the same value, i.e. f () =L if and only if!a!a f () =Land f () =L.!a +

3 Eamles 2!2 +3 2!! Theorem If!a f () =Aand!a g() =B both eist, then.!a (f ()+g()) =!a f ()+!a g() =A + B 2.!a (f () g()) =!a f ()!a g() =A B 3.!a (f ()g()) =!a f ()!a g() =A B 4.!a (f ()/g()) =!a f ()/!a g() =A/B (B 6= 0). In short: to take a it Ste : Can you just lug in? If so, do it. Ste 2: If not, is there some sort of algebraic maniulation (like cancellation) that can be done to fi the roblem? If so, do it. Then lug in. Ste 3: Learn some secial it to fi common roblems. (Later) If in doubt, grah it!

4 Eamles 2.! ! 2 = 0 because if f () = +3,thenf(2) = 0.!0!0 If f () = 2,thenf() isundefinedat =. However, so long as 6=, f () = 2 = ( + )( ) = +. So! 2 = +=+= 2.! 3.!0 +2 2!0, so again, f () isundefinedata. Eamles 3.!0 +2 2!0, so again, f () isundefinedata. Multily to and bottom by the conjugate:! =! =!0 ( +2+ 2) =!0 ( +2+ 2) = =! !! since (a b)(a + b) =a 2 b 2

5 Eamles.! ! 2 3.!0 4.!0 (3 + ) 2 3 2

6 Infinite its If f () gets arbitrarily large as! a, thenitdoesn thaveait. Sometimes, though, it s more useful to give more information. Eamle: For both f () = and f () =, 2!0 f () does not eist. However, they re both better behaved than that might imly:!0 +!0 =,!0 =!0 + does not eist!0 Why? A vertical asymtote occurs where f () =± and!a +!a =, 2!0 2 = f () =± 2 = Infinite its Badly behaved eamle: f () =csc(/) Zoom way in: (denser and denser vertical asymtotes) csc(/) does not eist, and!0 +!0 csc(/) does not eist

7 Limits at Infinity We say that a function f aroaches the it L as gets bigger and bigger (in the ositive or negative direction), written f () =L or f () =L!! if we can make f () as close to L as we want by taking su ciently large. (By large, we mean large in magnitude) Eamle :! =0 and! =0. Limits at Infinity: functions and their inverses Theorem If!a ± f () =, then! f () =a. If!a ± f () =, then! f () =a. Eamle: Let arctan() be the inverse function to tan(): y =tan(): -π/2 π/2 Since! /2 = and! /2 + = (restrict the domain to ( 2, 2 )tothatwecaninvert) y =arctan(): π/2 -π/2 we have! = /2 and! = /2

8 Rational functions Limits that look like they re going to can actually be doing lots of di erent things. To fi this, divide the to and bottom by the highest ower in the denominator! E!! 3 +2! Fi: multily the eression by /3 : / 3! 3 +2 =! 3 +2 / 3 / 3 =! = = 0 3 E 2 E ! 4 2!! Fi: multily the eression by /2 / ! 3 +3!! Fi: multily the eression by /3 / 3

9 Rational functions: quick trick! 3 +2 = ! 4 2 = ! 3 +3 = Suose P() =a n n + +a +a 0 and Q() =b m m + +b +b 0 are olynomials of degree n and m resectively. Then in general, 8 P()! Q() = a n n! b m m = a >< 0 n < m n n m = a n! b b m >: m n = m ± n < m Eamles: Other ratios with owers. Eamle:! [hint: multily by / / and remember a b = a 2 b.]

10 Evaluating its when! 0.. Show!0 ( 2 2) = Show!0 = Show!0 2 = 7/2. 4. Show! = 37/ Show! = 3/5. 6. Show h!0 7. Show!0 + h h = =/ Show =/(2 3).!0 9. Show = (/2) 3/2. h!0 h + h 2 0. Show =2 a.!0 a + a +. Show =/2.!0 2. Show = 2.! Show!0 e + e 2 2 =. 4. Show h!0 f( + h) h when f() = a + b. 5. Show h!0 f( + h) h when f() =(m + c) n. f() a = 2 a + b f() = mn(m + c) n

11 Evaluating its when! a.. Show! ( ) = 5. 5/2 a 5/2. Show!a a =(5/2)a 3/2. 2. Show! = Show = 2.! Show =.! Show! 3 = Show! =/2. ( + 2) 5/3 (a + 2) 5/3 2. Show!a a 3. Show! = Show! = 20/3. 5. Show! n = n. =(5/3)(a + 2) 2/ Show! 2 +2 = Show!a a a = 2 a. 8. Show! = Show = 325.!5 5 2 a 2 0. Show = 2a.!a a 3 7. Show =/2.!2 2 a Show = 2 3!a 3a Show!a n a n a = nan.

12 Evaluating its when!. +2. Show! 2 = Show! =3/ Show! 3 2 =/ Show! =2/7. (3 )(4 5) 5. Show = 2.! ( + 6)( 3) 6. Show! 42 + =/2. 7. Show! 2 = Show t! t + t 2 + = Show n2 + n = 0. n! 0. Show n2 + n n =/2. n!