0 Cotiuity Chapter 0: Cotiuity Subtopics.0 Itroductio (Revisio). Cotiuity of a Fuctio at a Poit. Discotiuity of a Fuctio. Types of Discotiuity.4 Algebra of Cotiuous Fuctios.5 Cotiuity i a Iterval.6 Cotiuity i the Domai of the Fuctio.7 Cotiuity of some Stadard Fuctios Type of Problems Eercise Q. Nos. Eamie the Cotiuity of Fuctio at a. Q. give poit Miscellaeous Q. (iii,v) Types of Discotiuity (Removable. Q. Discotiuity/Irremovable Miscellaeous Q. Discotiuity) Fid the Value of Fuctio at give poit if it is Cotiuous. Q.4 Fid Value of k/a/b/α/β if the Fuctio is Cotiuous at a Give Poit Eamie Cotiuity of a Fuctio over give Domai/Fid poits of Discotiuity/Show that give Fuctio is Cotious Fid the Value of k/a/b/α/β if the Fuctio is Cotiuous over a give Domai. Q.,5 Miscellaeous. Q., Q.(ii) Q.(v, vi, viii, i), Q. (i, ii, iii, iv, v, vii, viii, i, ) Miscellaeous Q. (iv), Q.4(i, iii), Q.5. Miscellaeous Q.(i, ii, iii, iv, vii), Q. (vi) Q.(i), Q.4 (ii)
Std. XII : Perfect Maths - II Itroductio (Revisio) I this chapter, we will discuss cotiuity of a fuctio which is closely related to the cocept of its. There are some fuctios for which graph is cotiuous while there are others for which this is ot the case. Limit of a fuctio: A fuctio f() is said to have a it l as teds to a if for every > 0, we ca fid a positive umber δ such that, f ( ) l < wheever 0 < a < δ, If a f () f (), a the the commo value is Algebra of its: a f (). If f() ad g() are ay two fuctios, i. [f() g()] f() g() iii. v. vii. i. a [f() g()] a a a [k.f()] k f ( ) a a a a a f() f ( ) [f()] g() f ( ) a a a g() f(), where k is a costat. Limits of Algebraic fuctios: i. a iii. a a g( ) a k k, where k is a costat. v. If P() is a polyomial, the Limits of Trigoometric fuctios: i. si 0 0 si iii. si 0 0 si v. si 0 80 vii. si k 0 si ta i. P() P(a) a ii. iv. vi. viii. ii. iv. vi. ii. iv. vi. [f() g()] a f ( ) a g( ) [f()] a a a a f() a f ( ) g( ), where f ( ) a log[f()] log f ( ) a a a a a 0 0 k viii.. r r a a a ta ta a 0 0 cos 0 si si( a) a a ta ta cos a 0 g() a ta ( a) a g() 0 a
Chapter 0: Cotiuity Limits of Epoetial fuctios: i. iii. a log a, (a > 0) ii. 0 ( ) e 0 iv. e 0 0 m a m log a log a m Limits of Logarithmic fuctios: i. iii. log ( ) 0 log ( ) 0 ii. Cotiuity of a fuctio at a poit Left Had Limit: log ( a ) 0 log a e, a > 0, a a f() deotes the it of f() whe approaches to a through values less tha a. L.H.L. a f() Right Had Limit: <a a f() f(a h), (h > 0).[Left had it] h 0 a f() deotes the it of f() whe approaches to a through values greater tha a. R.H.L. a f() a a > a a f () h 0 f() ad a f() are ot always equal. f() eists, if ad oly if a f(a h), (h > 0)...[Right had it] f() a f() i.e., L.H.L. R.H.L. Graphically, this ca be show as give i the adjoiig figure. Fuctio f is said to be cotiuous at a, if: i. f(a) eists ii. iii. iv. a f() eists a f() eists a f() a f() f(a) O Y y f() a f(a) X Discotiuity of a Fuctio f is said to be discotiuous at a, if it is ot cotiuous at a. The discotiuity may be due to ay of the followig reasos: i. ii. iii. c c c f() or c f() ad c f() ad c f() or both may ot eist. f() both eist but are ot equal. f() eist ad are equal but both may ot be equal to f(c).
Std. XII : Perfect Maths - II Cosider the fuctio defied by f() for for < Here, f() is defied at every poit i [, ]. Graph of this fuctio is as show adjacetly. Left had it at ad value of f() at are both equal to. But right had it at equals, which does ot coicides with the commo value of the left had it ad f(). Agai, we ca ot draw the cotiuous (without break) graph at. Hece, we say that the fuctio f() is ot cotiuous at. Here, we say that f() is discotiuous at ad is the poit of discotiuity. Types of Discotiuity: i. Removable discotiuity: A real valued fuctio f is said to have a removable discotiuity at c i its domai, if 4 f() f(c) c i.e., if f() c f() f(c) c Eg. Cosider the followig fuctio, 6 f(), 4 4 5, 4 Here f (4) 5 6 ( 4)( 4) f() 4 4 4 4 4 4 f is discotiuous at 4. ii. ( ) 4 4 4 8 f(4) f() eists but c Now, let us fid why f() is discotiuous at 4. I the above fuctio f() eist, but is ot equal to f(4) sice f(4) 5. 4 This value of f (4) is just arbitrarily defied. Suppose, we redefie f() as follows: 6 f(), 4 4 8, 4 The f() becomes cotiuous at 4. The discotiuity of f has bee removed by redefiig the fuctio suitably. Note that we have ot appreciably chaged the fuctio but redefied it by chagig its value at oe poit oly. Such a discotiuity is called a removable discotiuity. This type of discotiuity ca be removed by redefiig fuctio f() at c such that f(c) f(). c Irremovable discotiuity: A real valued fuctio f is said to have a irremovable discotiuity at c i its domai, if f() f() c c i.e., if f() does ot eist. c Such fuctio caot be redefied to make it cotiuous. X Y 0 Y 4 X
Chapter 0: Cotiuity Eercise.. Eamie the cotiuity of the followig fuctios at give poits 5 e e i. f() si, for 0, for 0 log00 log(0.0 ) ii. f(), for 0 00, for 0 iii. f(), for, for log log7 iv. f(), 7 for 7 7, for 7 v. f() ( ), for 0 e, for 0 at 0 at 0 at at 7 at 0 vi. f() 0 7 4 5 cos 4 0, 7, for 0 vii. f() si cos, for 0, for 0 at 0 for 0 [Oct ] at 0 [Mar 4] log( ) log( ) viii. f(), ta for 0, for 0 si i. f(), for, for. f(), for 0 c, for 0 at 0 at at 0 (where c is arbitrary costat) [Oct 5] i. f(), for 0 <, for < at 5
Std. XII : Perfect Maths - II ii. 9 f(), for 0 < <, for < 6 9, for 6 < 9 si iii. f() cos, for 0 < cos, for < < e iv. f(), for 0 e, for 0 at ad 6 at at 0 6 v. f() 5... ( ) ( )(4 ) 6 ( ) Solutio: i. f(0).(give) f() 0 0 5 e e si 0, for, for e ( e ) si e e e (0) 0 si e 0 f(0) Sice, f() f(0), f is cotiuous at 0. 0 ii. f(0) 00 f() 0 Sice, 0 0 00.(give) ( ) log00 log 0.0 log 00 0 0 00 0 00 ( ) 00 f(0) 0 ( ) log 00 ( ) 00 0 f() f(0), f is cotiuous at 0.. ( ) log 00 e si Q, 0 0 ( ) at log. Q 0
iii. f().(give) f() a a a Sice, f() f(), f is discotiuous at. (). Q ( a) iv. f(7) 7.(give) log log 7 f() 7 7 7 Put 7 h, the 7 h, as 7, h 0 h 7 log log (h 7) log 7 7 f() 7 h 0 h h 0 h h h log 7 h 0 h 7 log 7 7 0 h 7 7 Sice, 7 f(7) f() f(7), f is discotiuous at 7. 7 v. f(0) e.(give) f() 0 0 ( ) ( ) 0 0 ( ) e. ( ) Q e 0 f(0) Sice, f() f(0), f is cotiuous at 0. 0 7 (). log ( ) Q 0 Chapter 0: Cotiuity vi. f(0) 0 7...(give) f() 0 0 0 0 0 7 4 5 cos 4 ( )( 5 7 ) ( ) si.5 5 7. 7 si 0 0 5 7 4 si 4 ( )( 5 7 ) si 0 ( ) ( ) 5 7 si [ Mark] 7
Std. XII : Perfect Maths - II 8 Sice, 0 5 7 0 0 0 si 8 0 ( ) log log5 log 7 8() 5 log log 7 f(0) 8 f() f(0), f is discotiuous at 0. vii. f(0).(give) f() (si cos ) si 0 Sice, viii. f(0) 0 0 f() 0 0 0 f() f(0), f is cotiuous at 0. 0 0 0 0.(give) log log ( ) ( ) ta log ta log ta log log ta log log 0 ta log log 0 0 ( ) ta 0 log log 0 0 cos si 0 cos 0 0 f(0) [ Mark] [ Mark] [ Mark] [ Mark]
Chapter 0: Cotiuity Sice, () (). log( ) 0 f() f(0), f is cotiuous at 0. 0 i. f.(give) f() si Put h, the as, h 0 si h f() cos h h 0 h h 0 h Sice, h si 4 h 4 h h 4 si h si f f() f, f is discotiuous at. h [ Mark] () [ Mark] [ Mark]. f().(give) Thus,, if 0, if 0 f(), if 0, if 0 i. Now, ad Sice, f() 0 0 f() 0 () 0 () f() f(), f is discotiuous at 0. 0 0 f f() Sice, f() ( ) f() f() f, f is cotiuous at. 9
Std. XII : Perfect Maths - II ii. Case : Whe, f() 6 0 f() Sice, f() f() 9 ( ) 6 ( ) 6 f() f(), f is cotiuous at. Case : Whe 6, f(6) 9 6 f() ( ) 6 9 6 6 9 f() 6 6 ( ) 6 6 Sice, f() f() f(6), f is discotiuous at 6. 6 6 si iii. f si si 0 0 cos cos ( ) 0 cos si f() cos si cos si cos (0) 0 f() cos Put h, the as, h 0 cos h f() si h si h h 0 h h 0 h h si h h () Sice, f() f() f, f is discotiuous at. iv. f(0) f() 0 0 e e...(give) As 0,, thus e e e i.e., e i.e., e 0 0 e 0 0 e Also, as 0,, thus e e i.e., e i.e., 0 e
Chapter 0: Cotiuity 0 Sice, e 0 0 e f() f(), f is discotiuous at 0. e e 0 0 ( )(4 ) v. f().(give) 6 5... ( ) f() Sice, ( ) 5... ( ) 5... ( ) ( ) ( ) ( ) 5( )... ( )( ) ( ) 5... ( ) ( ) 5 ( ) ( )... ( ) () 5()... ( )(). Q ( r ) r ( r r) r r 6 r r r ( )( ) ( ) ( ) ( ) ( ) ( 4 ) ( )( 4 ) 6 f() f(), f is cotiuous at.. Fid the value of k, so that the fuctio f() is cotiuous at the idicated poit ( k e ) i. f() r si k, for 0 4, for 0 at 0. Q ( ) r r r ii. f(), si for 0 k, for 0 iii. f(), for k, for iv. f(), for 0 k, for < 0 at 0 at at 0
Std. XII : Perfect Maths - II v. cos 4 f(), for < 0 k, for 0, for > 0 6 4 vi. f() log( k ), si for 0 5, for 0 vii. f() 8, k for 0, for 0 viii. f() k ( ), for 0 4, for > 0 cot i. f() (sec ), for 0 k, for 0 at 0 at 0 at 0 at 0 at 0 ta. f(), for k, for Solutio: i. f(0) 4.(give) Sice, f() is cotiuous at 0 ( k ) k e si k e si k f(0) f() k k 0 0 k k e si k 0 k k 0 k 0 k k ()() k f(0) k 4 k k ± ii. f(0) k. (give) Sice, f() is cotiuous at 0 f(0) f() 0 0 si 0 si k log log log 9 0 si 0 0 0 si 0 ( ) ( ) si (log) (log ) iii. f() k.(give) Sice, f() is cotiuous at f() ( ) 0 f() 0 Sice, f() f(), f is cotiuous at. at log
Chapter 0: Cotiuity k 0 Now, f() f() 0 iv. Sice, f() is cotiuous at 0 f() f() 0 0 ( k ) 0 0 k 0 k k 0 ( ) v. f(0) k.(give) Sice, f() is cotiuous at 0 f() f() f(0) 0 0 Cosider, f(0) f(0) 8 k 8 cos4 f() 0 0 si 0 4 4 8 0 si 0 si si 8 4 0 vi. f(0) 5.(give) Sice, f() is cotiuous at 0 log ( k) k.log ( k) log ( k) f(0) f() 0 k 0 0 si 0 si si 0 f(0) k k 5 vii. f(0).(give) Sice, f() is cotiuous at 0 f(0) 8 f() 0 0 k 0 8 0 k 0 8 8 0 0 k k 0 log8 log log k log log k log log k k 8 log log 4 log k log k log log k log log k ( 8 ) ( ) k log ( k) k 0 k k si k 0
Std. XII : Perfect Maths - II viii. Sice, f is cotiuous at 0 f() f() 0 0 0 k( ) k(0 ) 4(0) k k 0 4 i. f(0) k.(give) Sice, f() is cotiuous at 0 f(0) f() 0 ( cot sec ) 0 ( ta ) 0 ta k e. f k.(give) Sice, f() is cotiuous at e e. ( ) Q 0 f f() ta Put h, the, as, h 0 ta h h 0 h h 0 h 0 ( ta h) ( ta h) h ( ta h) ( ) h 0 ta ta h ta ta h h h 0 4 ta h ta h 4 h h 0 h ( ta h) ta h ta h ta h h ta h ta h ta h h h ( ta h ) h 0 4 ( 0) 4 ta h () 4 k 4. Discuss the cotiuity of the followig fuctios, which of these fuctios have a removable discotiuity? Redefie the fuctio so as to remove the discotiuity. 4 ( ) i. si f(), for 0, for 0 ii. f() cos ta, for 0 9, for 0 at 0 at 0
iii. f() ( e ) ta, for 0 si e, for 0 si iv. f(), for ( ) 7, for at 0 at Chapter 0: Cotiuity v. f() 4 e, 6 for 0 log, for 0 ( e ) vi. f() si, for 0, 60 for 0 at 0 [Mar 6] at 0 vii. f() (8 ), i [, ] {0}; si log 4 Defie f() i [, ] so that it becomes cotiuous at 0. viii. f(), for <, for Solutio: i. f(0).(give) f() 0 0 0 f() f(0) si( ) si( ) si( ) ( ) ( ) (0 ) 0 ( ) 0 ( ) f is discotiuous at 0. The discotiuity of f is removable ad it ca be made cotiuous by redefiig the fuctio as ( ) si f(), for 0, for 0 ii. f(0) 9.(give) cos f() 0 0 ta 0 0 9 si 4 9 4 ta si ta 0 si 9 0 ta 0 at 0 si ta 9 ( ) 9 at 5
Std. XII : Perfect Maths - II 0 f() f(0) f is discotiuous at 0. The discotiuity of f is removable ad it ca be made cotiuous by redefiig the fuctio as f() cos ta, for 0 9, for 0 iii. f(0) e.(give) f() 0 f() 0 f() f(0) 0 (e ). ta (e ) ta 0 si 0 si (e ) ta e ta 0 0 0 si si 0 f is discotiuous at 0. The discotiuity of f is removable ad it ca be made cotiuous by redefiig the fuctio as ( e ) f() ta for 0 si for 0 at 0 at 0 iv. f 7 f() ( ).(give) si si Put h, the h As, h 0 si h cos h f() h 4h h 0 f() f ( ) si h 4h ( cos h) si h 4 h cos h 4h ( cos h) cosh cosh h 0 4 () cos h 4h ( cosh) 8 f is discotiuous at. 6
Chapter 0: Cotiuity The discotiuity of f is removable ad it ca be made cotiuous by redefiig the fuctio as si f(), for ( ) 8, for at v. f(0) log f() 0 4 e 0 0 0 6 (4 ) (e ) 6 4 e 6 log 4 log e log 6 4 log e log 6.(give) 4 e 0 6 4 e 0 0 6 0 f() f(0) 0 f is discotiuous at 0. The discotiuity of f is removable ad it ca be made cotiuous by redefiig the fuctio as f() 4 e, for 0 6 4 log e, for 0 log 6 at 0 [ Mark] [ Mark] [ Mark] [ Mark] vi. f(0) 60 f() 0 ( e ) 0...(give) si e si 0 si e. 80 si e 0 80. 80 0 80 0 80 80 f() 0 80 60 Sice, f() f(0), f is cotiuous at 0. 0 log e () 80 7