LIMITS OF FUNCTIONS (I)

Transcription

1 LIMITS OF FUNCTIO (I ELEMENTARY FUNCTIO: (Elemeary fucios are NOT piecewise fucios Cosa Fucios: f(x k, where k R Polyomials: f(x a + a x + a x + a x + + a x, where a, a,..., a R Raioal Fucios: f(x P (x, where P, Q are polyomials. Q(x Expoeials: f {e x, b x }, where b R Logarihms: f {l x, log x, log b x}, where b R Trig Fucios: f {si x, cos x, a x, csc x, sec x, co x} Iverse Trig: f {arcsi x, arccos x, arca x, arccsc x, arcsec x, arcco x} Ay fiie sum, differece, produc, quoie, power, roo, or composiio of ay of he above elemeary fucios. PIECEWISE FUNCTIO: A piecewise fucio has more ha oe fucio (piece i is defiiio, spaed by a sigle large lef brace. { { { x, if x < x, if x (, x, if x < e.g. f(x + x, if x + x, if x [, + x, oherwise f(5 + 5, f( +, f( ( 7 { { { { x, if x x, if x [, x, if x e.g. x x, if x < R+ {} x, if x x, if x (, x, if x R x, oherwise,, 7 ( 7 7 +, if < π/ +, if (, π/ +, if < π/ si if π/ < π si if [ π/, π si if π/ < π e.g. g(, if π, if {π}, if π log if > π log if (π, log oherwise g( 8 + ( 8 5, g ( ( π 4 si π 4, g(π, g( log log INTERESTING PROPERTIES OF INFINITY: Remember, is o a real umber, bu raher a symbol idicaig growh wihou boud. Similarly, idicaes decay wihou boud. However, eve hough ± are symbols, hey saisfy some arihmeic properies ha agree wih iuiio: (E. + (E. x R, + x x + ad + x x (E. ( (, ( (, ( ( ( ( (E.4 x > x ad x (, x < x ad x ( (E.5 N ad INDETERMINANT FORMS:,,,,,, A his poi (i.e. early i he course, oly he ideermia forms ad will be ecouered. c Josh Egwer Revised Sepember,

2 LIMIT DEFINITIO/NOTATION: The value of f(x a x c is y f(c y (-sided i The i of f(x as x approaches c is L f(x L f(x L as x c (-sided i The i of f(x as x approaches c from he lef (egaive side is L f(x L (-sided i The i of f(x as x approaches c from he righ (posiive side is L f(x L + (ifiie i f icreases w/o boud as x approaches c f blows up as x approaches c f(x + (ifiie i f decreases w/o boud as x approaches c f ose-dives as x approaches c f(x REMARK: Whe x approaches c, i oly makes sese ha x be sufficiely close o c. RELATIOHIP BETWEEN -SIDED & -SIDED LIMITS: LIMIT RULES: DNE meas Does No Exis f(x L f(x f(x L + (L. (Cosa Rule k k, where k R (L. (Muliple Rule [kf(x] k f(x, where k R (L. (Sum/Diff Rule [f(x ± g(x] f(x ± g(x [ ] [ ] (L. (Produc Rule [f(xg(x] f(x g(x f(x f(x (L.4 (Quoie Rule, provided g(x g(x g(x [ ], (L.5 (Power Rule [f(x] f(x where Q ad f(x exiss (L. (DNE Rule Afer simplificaio, i of par of a fucio is DNE i of eire fucio is DNE. SPECIAL LIMITS: (S. + x +, N + x (S. + eve N x si x (S. x cos x x { +, if N is eve x, if N is odd x Le A {, 5π, π, π, π, π, 5π, }. The: a x + x A Le B {..., π, π, π,, π, π, π,... }. The: co x x B x DNE odd N l x log x log b x a x x A + co x + x B + ALGEBRAIC PROCEDURE FOR LIMITS OF ELEMENTARY FUNCTIO:, odd N x sec x DNE x A csc x DNE x B Firs, compue he -sided i: f(x If f(x L, he f(x f(x L. Else, f(x DNE -sided is mus be formally compued. + ( (aïve subsiuio If f(c is defied, ha is, f(c L R, he f(x f(x f(x L + (ideermia If f(c or, rewrie ad/or simplify f(x firs, he apply aïve subsiuio o ge he aswer. Simplificaios: Raioalize umeraor/deomiaor, combie fracios, facor polyomial(s, mold io special i,... (CV (rasformaio Someimes o mold a i io oe of he special is, a chage of variables is ecessary. If f(c a, a, he use i rules, simplificaio, ad/or rasformaio o reduce o a i of x or x (rig is For is of rig, someimes i s bes o rewrie rig expressio i erms of si( ad cos( firs. ALGEBRAIC PROCEDURE FOR LIMITS OF PIECEWISE FUNCTIO: Firs, compue he -sided is: If f(x f(x ad f(x + f(x L, he he -sided i f(x L. + Else, f(x DNE c Josh Egwer Revised Sepember,

3 : x (x x, Sice he fucio is elemeary, fid he -sided i firs. x, ad x x ( (x x ( +(x x (x x ( ( x (x x x x x ( (x x ( +(x w w w w si 7π/ w w si ( si 7π 7π/ ( (. w ( 7π Y 7Y + Y Y / 5Y + Y 7 4 7π Y 7Y + Y Y / 5Y + Y 7 Y 7Y + Y Here, facor umeraor & deomiaor: 5Y + Y 7 Now, simplify & ry aïve subsiuio agai: 4 7π 4 w w ( 7 ( + ( 5 ( + ( 7 7π si. 4π 7π/ 4π 7Y (Y + (Y (5Y + 7(Y Y 7Y + Y (7Y + (Y Y / 5Y + Y 7 Y / (5Y + 7(Y 7Y + Y / 5Y + 7 Y 7Y + Y 8 Y / 5Y + Y ( + 5 ( α α ( + α Sice he fucio is elemeary, fid he -sided i firs. + α + ( α α ( rewrie/simplify fucio!! Here, combie erms of umeraor io oe fracio & facor deomiaor: Now, simplify & ry aïve subsiuio agai: + α α α α + α α α α+ α (α + (α α + α α α ( + α α (α + (α α α ( + + α α rewrie/simplify fucio!! (7Y + (Y (5Y + 7(Y α α α + α+ α α α α (α + (α α(α ( [( ] ( 4 ( rewrie/simplify fucio!! Here, raioalize umeraor: ( 4 ( ( ( 4 ( ( 4 Now, simplify & ry aïve subsiuio agai: ( ( ( 8 8 c Josh Egwer Revised Sepember,

4 : csc, csc, ad csc + Sice he fucio is elemeary, fid he -sided i firs. Firs, wrie fucio i erms of, cos : csc csc si reduce i o oe ivolvig or, where N Mold fucio io oe ivolvig : csc csc ( L. ( L.4 ( S. [ ] ( ( Now, sice he -sided i DNE, fid he -sided is formally (recyclig work already performed: ( [ ] ( csc S. S. ( + csc + ( x + [ ] ( S. S. + ( + + : x + 4x, x + 4x, ad Sice he fucio is elemeary, fid he -sided i firs. x x + 4x ( + 4( x x + x + 4x S. DNE reduce i o x, where N ( ( ( Facor fucio: x + 4x (x + x 5 (x (x + 5 x x + 5 ( ( ( ( ( ( ( ( x x + 4x L. L. x x x + 5 x x x x x + 5 x ( ( ( ( + 5 x x, which is very close i form o, bu o quie! x x x Hece, use a rasformaio (AKA chage of variables: Le u x. The, x u +, which meas x (u + u So, x x + 4x CV x x S. L. (DNE DNE u u Now, sice he -sided i DNE, fid he -sided is formally (recyclig work already performed: x x + 4x CV x x S. E.4 ( u u x + x + 4x x + x + π, ( π CV u, ad + π u + S. ( π + Sice he fucio is elemeary, fid he -sided i firs. + π + π ( π + π E.4 (+ + + π reduce i o, where N, which is very close i form o + π, bu o quie! Hece, use a rasformaio (AKA chage of variables: Le u + π. The, u π, which meas π (u π π u So, CV S. + π u (DNE L. DNE u Now, sice he -sided i DNE, fid he -sided is formally (recyclig work already performed: CV S. ( π + π ( E.4 + u u CV S. ( π + + π (+ E.4 u + u c Josh Egwer Revised Sepember, ( x + 5 x x 4

5 : Z 7 (Z 7, Z 7 (Z, ad 7 Z 7 + (Z 7 Sice he fucio is elemeary, fid he -sided i firs. Z 7 (Z reduce i o 7 Z Z, where N The i is very close i form o, bu o quie! Z Hece, use a rasformaio (AKA chage of variables: Z Le u Z 7. The, Z u + 7, which meas Z 7 (u u So, Z 7 (Z CV S. + 7 u u Now, sice he -sided i exiss, he -sided is share he same value as he -sided i: Z 7 (Z 7 (Z 7 + EXAMPLE: Le f(x (a f si Z 7 + x 4, if x < π/ si x, if x π/ (b Sice f is piecewise, fid he -sided is firs. f(x x π/ 4 ( π 4 8 sice f(x x π/ 8 π 4 f(x, i follows ha x π/ +, if <, if < EXAMPLE: Le g( 5, if, if < 4 +, if > 4 (a g( 5 (b Sice g is piecewise, fid he -sided is firs. g( ( Evaluae: (a f(π/ (b x π/ f(x f(x si x π/ + g( ( + sice g( g(, i follows ha g( + f(x DNE x π/ Evaluae: (a g( (b g( c Josh Egwer Revised Sepember, 5