Math Analysis I - Limits of Functions Part 1 (Functions)

Composing with results in a translation of the graph

LR#IR,to: (x)t a - x t ct afComposing with results in a translation of the graph off :in p a r t i c u l a r,(x) (Xto) H O R I Z O N TA L1 f o ta T R A N S L AT I O Nj= thet oC e o rightt h et o left( y ojCx1fCxtc1cxo-f@JG4ta.ceoT R A N S L AT I O NVERTICAL2- t o o (x) (x)f f= t o C e o downwards^¥. upwardsc y oMICHf e l t c t oc ,Ta t e . ⇐2. Math Analysis Limits of Functions 16 of 36Resceeingl e t r e a l functionI b e a I R I {o}fix s u c h t h a tECR#R, ( x )e e : n - e xC C f :fThe composition of with has the effect of rescaling the graph ofK(x) (ex)- ff o r e1 towards YO Nif XCOMPRESSIONflex)1,C y• fif O NSTRETCHING(ex)i c e X1 ,0• yaway from#¥:Y- f l e x ) , c > ,fat - s e n o r a ,(x) (x)ro o f2- - e t Xaway fromif O N YSTRETCHINGeffx)I ,C t• O Ne fif Y(x)i c e COMPRESSION1,0• towards Xy-etch, c > ,np#¥%"'""2. Math Analysis Limits of Functions 17 of 36Reseectionl e t functionr e a lI be ax)I f the elong Y1 of

(jar)reflectionis#¥fl-x) Ice)11×1) R t # R2 - YI the ofreflectionis f: o n¥ ¥111×1) fix)I f I(x) X3- flawt h eis re f l e c t i o n of o n# ¥11h11far)2. Math Analysis Limits of Functions 18 of 36

Elementary functions

Elementary functions play an important role in mathematical analysis since the majority of the applications can be derived from them trough algebraic operations, composition and inversion.

Definition ( e v e u . e e # c t i o n s )IRE#R, E bel e t withcj : a m a pdomain t osymmetric t ow i t h respectathe Es u c horigin EX ,t h a t - X Vx EEffx)1 if -ya)s a i d1 is EVEN Ysymmetric w i t h t orespectst§µ, jarl-ICHf r e ef f x )if =-far)2- S e i dI is ODDsymmetric w i t h Ot orespect*¥¥-sixteen2. Math Analysis Limits of Functions 19 of 36

Definition (periodicity)IRE # R ,l e t E E Ef: Xc , Rt)( I E1saidf periodPEMODIC ofis (i.t rI E E Eif E E , invarianti t ise . treslations)underI) t xj e t E Efar)t h esuch = ,let {03,N iK E t h e n

1. 1KI)(x periodicf periodis o ft I saids u c h fatt h a t PERIODMINIMUMi s o fperiodicc o s t e n t is anya o fm a p period)(thereI >period n o m i n i m u miso2.
2. Math Analysis Limits of Functions 20 of 36
3. Power f u n c t i o n s y=×4
4. These are functions of the formi) is a-o, y-l {o}a) IN y-x" IRIRli f c- n iuL - u , j:-C-x)"I x" EVENish e v e n IR R # R t× " t h e j ,I o Xb x " × 2*±.then IRIRf : → is n o t injectivehowever, b y r e s t r i c t i o nR t # R t Rtf: injective invertible E Eis i n(-x)" = - X "odd2- is ODDuxIo IR'Rt, th 112nsFx IRf:x e o Ee aHt .3×5×7X2.
5. Math Analysis Limits of Functions 21 of 36
6. iii) g e n e r a l i s a t i o n s2 1 6 3 x"if a - n , en ye(previous case)1Nif t h e no , xsu E• Zx-n,if t h e ni Eo ,n ny =• = L INy n E,× " x-h{03112403 R if :t h e n , jarl↳ =,1 is e v e nn f e x - 2(x)f .2- oddisu #¥÷.2.
7. Math Analysis Limits of Functions 22

of 36id - T xx "t u t {03Z ia -Yu ,if V m( x may=(×")'m Yu - ×× n.since =x" whenever× "is the inverse o f× " invertibleis1 isn e v e n Rtbegone E invertiblej :a s is→Rt byEif C restrictionx'" " Tx Rt)( E cI R ,then f: R ( x )f, =→ -⇐¥€?N=xk-rxodd2- ish alwaysthen invertiblex " is x" "ifI R , j'IR1 : (xu)(x)→ f == -^,¥¥¥. A i r# '2. Math Analysis Limits of Functions 23 of 36↳ 11240thatif irretionee s u c h2 t o i s a e. too)[ o,set X - e t h a tsuch a Ethen, b y d e n s i t y nationalso f { a m I ex}ed-sup " i n1 if e z 1 , {a"'m 14ms a}el-ing2- i f e e l ,o r least)[o, t a ) ( a tx" definedthen is o n INJECTIVE'K IRHt h e alwaysinverse of is L Ex ,2. Math Analysis Limits of Functions 24 of 36Exponential functionslet R is constantt h a te ± o Es u c h a Rf xt h e n y-ex defined Eis IRt xif l - l Ey-l• ,if S T

R I C T LY INCREASING1 ,e• if DECREASINGf e e l S T R I CT LY0•R e m e r : ext'tx, -exitIR,y e • e -Y- E yX• 'sexy(E)• - ×IR#RT (te){03, text-ex-f : (f) ×t h e n £REFLECTIONe- × WITHO F= YRESPECT O Fo¥¥¥× e xa [a> D2. Math Analysis Limits of Functions 25 of 36Logarithmic functionsTCH-e" inactive#invertiblelet isthene t I, j ' -logexe t(x) (x)f LOGARITHM=R t l fo} IRf: ya-logex→ , e l a te "e t - x y-logex - x,Re-mezk: logel-o. to-I][log.Coger-lux bye--I loge - l i n e - 1 ,= .- log= x b o g e y,' Te n t - l a t e x tx,+ y oy( •" n a t u r a l Ty-logex-logeylogarithm) tryb y e s o. Fye 112XY= F x >l a t e xlog,_ byex- Lo g x y o,.- LOT I Fxloge X t o=• lob 0~1¥..........( e x i tlogex.2. Math Analysis Limits of Functions 26 of 36Trigonometric functions1122 -Denote here by X, Y the coordinates on the Cartesian plane , and considerthe unit circle, i.e., the

circle of unit radius centred at the origin O = (0, 0), whose equation reads: x2 + y2 = 1.

Starting from the point A = (1, 0), intersection of the circle with the positive x-axis, we go around the circle. More precisely, given any real x we denote by P(x) the point on the circle reached by turning counter-clockwise along an arc of length x if x ≥ 0, or clockwise by an arc of length -x if x < 0.

The point P(x) determines an angle in the plane with vertex O and delimited by the outbound rays from O through the points A and P(x) respectively.

The number x represents the measure of the angle in radians.

Oleg (gned¥¥I¥¥¥¥t¥Oleg 3 0 4 5 180 360G o T o 1 2 0 270135 150&§-32¥A y) -Cotton)µ#¥a ( x . (cosPcx) s i n x )- x .-asx."%Q (x)• ⇐X- 8 ¥ - o n - I s i n x es i n I¢u× aux-ftp.oit-ieosxei, then, X - o x , Y - s i n xFUNDAMENTAL GONIOMETRIC IDENTITYWEIRs i n e -1cos2x→ t .X-Oke S t a xt e n UNBOUNDED= -OH c o s x2.

Math Analysis Limits of Functions 27 of 36

Increasing or decreasing by 2π the length x has the effect of going around the circle once, counter-clockwise or clockwise respectively, and returning to the initial point P(x). In other words, there is a periodicity:

2T) R (R-21T)(x,P = p t x( x ) ,I E EE2T)then, (or K2kt,a t& - L t IR)(fire2kt)(x2kt)(x - G s xtts i n - s i n c o s× ,Hence, IR#I-i. R#Eti'tII,s i n : cos:÷:÷#¥¥÷±÷÷IR( x It-cos t x tthen, s i n + x X-Ft/"" K i tX - K i ts i n - O,X- O, X-zit X - 2 k t2ktSiu cosx - l , l,x =X - - Isin ⇐ c o s t× --I X - T t 2kt=-l,i x .I = ( 2 k t 1)i t2.

Math Analysis Limits of Functions 28 of 36

The tangent function y = tan x (sometimes y = tg x) and the cotangent function y = cot x (also y = ctg x) are defined by

8 1 1, I t h eten 2t Kit,t o fxx c o s tx= kit)t I zitIzthen d o t e n KI,t =i n = ,IR Z}{x-I K Ei KI,t= Zs%¥, thet e k i t ,a t Efsix xo =ru=

l-KitUt h e n Kit)do c o tm =v= ZK E {x-KI,IR Z}K Ei=R i Ix-Izten: RZ}KI, K Et →Z} RR t {x-KI, →K Ec o t : ÷t¥¥¥÷nl ll l ll lI lI lll ll l2. Math Analysis Limits of Functions 29 of 36

Remarks (trigonometric f o r m u l a s )(dtp)1 Posacop sins i n ±s i n= a DUPUCATION2s i n s i n2 x = x 6 s x(http) cop sinasinpcos cos Ia= c o s t - s i n k DUPUCATIONcos w e 2 6 5 221 - 2 5 i n X - 1= x =- t e n a t t e n p yt a n k ±p) t e n a t e u pI FZ t e(za)t e n D U P U C AT I O N= l-tonk XII×±¥2- Is i n sing coss i n× 2= ¥4XIIs i n6 5 = - 2X-cosy s i n x - YXty2Cost c