J-Mathematics LIMIT. INTRODUCTION : The cocept of it of a fuctio is oe of the fudametal ideas that distiguishes calculus from algebra ad trigoometr. We use its to describe the wa a fuctio f varies. Some fuctios var cotiuousl; small chages i produce ol small chages i f(). Other fuctios ca have values that jump or var erraticall. We also use its to defie taget lies to graphs of fuctios. This geometric applicatio leads at oce to the importat cocept of derivative of a fuctio.. DFINITION : Let f() be defied o a ope iterval about a ecept possibl at a itself. If f() gets arbitraril close to L (a fiite umber) for all sufficietl close to a we sa that f() approaches the it L as approaches a ad we write f() L ad sa the it of f(), as approaches a, equals L. a This implies if we ca make the value of f() arbitraril close to L (as close to L as we like) b takig to be sufficietl close to a (o either side of a) but ot equal to a.. LFT HAND LIMIT AND RIGHT HAND LIMIT OF A FUNCTION : Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65 The value to which f() approaches, as teds to a from the left had side ( a ) is called left had it of f() at a. Smbolicall, LHL f() a h 0 f(a h). The value to which f() approaches, as teds to a from the right had side ( a + ) is called right had it of f() at a. Smbolicall, RHL f() a h 0 f(a + h). it of a fuctio f() is said to eist as, a whe f() f() Fiite quatit. ample : Graph of f() 0 Fig. Importat ote : I f ( ) a, a ecessaril implies a a f() f( h) f( ) 0 h0 f() f(0 h) f(0 ) 0 0 h0 f() f(0 h) f(0 ) 0 h0 f() f( h) f( ) h0 f() f( h) f( ) 0 h0 f() f( h) f( ) 0 h0 f() 0 ad f() _ does ot eist. a. That is while evaluatig it at a, we are ot cocered with the value of the fuctio at a. I fact the fuctio ma or ma ot be defied at a. Also it is ecessar to ote that if f() is defied ol o oe side of a, oe sided its are good eough to establish the eistece of its, & if f() is defied o either side of a both sided its are to be cosidered. As i cos 0, though f() is ot defied for >, eve i it s immediate viciit.
J-Mathematics Illustratio : Cosider the adjacet graph of ƒ () Fid the followig : 4 (a) ƒ() 0 (b) ƒ() 0 (c) ƒ() (d) (g) ƒ() ƒ() (e) (h) ƒ() ƒ() (f) (i) ƒ() ƒ() 4 0 4 5 6 (j) ƒ() 4 (k) ƒ() (l) ƒ() Solutio : (a) As 0 : it does ot eist (the fuctio is ot defied to the left of 0) 6 (b) As 0 + : ƒ () ƒ(). (c) As : ƒ () 0 ƒ(). (d) As + : ƒ () ƒ(). (e) As : ƒ () ƒ(). (f) As + : ƒ () ƒ(). (g) As : ƒ () ƒ(). (h) As + : ƒ () ƒ(). (i) As 4 : ƒ () 4 ƒ() 4. 4 (j) As 4 + : ƒ () 4 ƒ() 4 4. (k) As : ƒ () ƒ(). (l) As 6,ƒ () ƒ() it does ot eist because it is ot fiite. 6 Do ourself - : (i) Which of the followig statemets about the fuctio () graphed here are true, ad which are false? (a) f () (b) f () does ot eist () (c) f () (e) (g) (h) (i) (d) f () does ot eist (f) f () eists at ever c (, ) c f () eists at ever c (, ) c f () 0 (j) f () f () f () 0 0 4. FUNDA MNTAL THORMS ON LIMITS : b g f () does ot eist. Let f l & g m. If l & m eists fiitel the : a a b g ( a ) Sum rule : f g m a b g b g 0 l ( b ) Differece rule : ( c ) Product rule : f. g l. m ( d ) Quotiet rule : f a a g f g l m a b g b g l, provided m 0 m Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65
(e) Costat multiple rule : kf k f a a ; where k is costat. ( f ) Power rule : If m ad are itegers the m / m / ( g ) f g f F H g I K f m a a f() a b g b g b g ; provided f() is cotiuous at m. J-Mathematics m / l provided l is a real umber. For eample : (g()) [ g()] a a (m); provided is cotiuous at m, m g(). a 5. INDTRMINAT FORMS : 0 0,,, 0,, 0 0, 0. Iitiall we will deal with first five forms ol ad the other two forms will come up after we have goe through differetiatio. Note : (i) Here 0, are ot eact, ifact both are aproachig to their correspodig values. (ii) We caot plot o the paper. Ifiit ( ) is a smbol & ot a umber It does ot obe the laws of elemetar algebra, (a) (b) (c) (d) 0 0 6. GNR AL MTHODS TO B USD TO VALUAT LIMITS : ( a ) Factorizatio : Importat factors : (i) (ii) a ( a)( + a +... + a ), N + a ( + a)( a +... + a ), is a odd atural umber. Note : a a a a ( ) Illustratio : valuate : Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65 Solutio : Illustratio : Solutio : We have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 6 ( ) ( ) ( )( ) ( ) ( ) ( ) 6 / is equal to (A) 8 (B) 4 (C) (D) oe of these 6. 6 / /. 4 8 6. 8 4 / / A s. As. (A)
J-Mathematics Illustratio 4 : valuate : P (P ) P ( ) Solutio : P (P ) P ( ) P P P P ( ) P( ) ( ) ( ) Dividig umerator ad deomiator b ( ), we get P ( ) P ( ) P (... ) P ( ) 0 form 0 P (... ) (...upto P times) ( ) P ( ) ( ) ( ) ( )... ( ) ( ) ( ) ( ) + () + () +...+ P() P + + +...+ P Do ourself - : (i) valuate : 7 5 P(P ) A s. ( b ) Ratioalizatio or double ratioalizatio : 4 5 Illustratio 5 : valuate : Solutio : 4 5 (4 5 )( )(4 5 ) ( )(4 5 )( ) (5 5) ( ) 4 5 5 8 0 Illustratio 6 : valuate : 5 Solutio : This is of the form 0 if we put 0 To eiate the 0 0 it ( 4 A s. factor, multipl b the cojugate of umerator ad the cojugate of the deomiator 8 5 8 0 0 ) ( 8 0 ) ( 8 0 ) ( 8) (0 ) ( ) (5 ) ( 5 ) ( 5 )( 5 ) 5 8 0 As. Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65
J-Mathematics Do ourself - : p p (i) valuate : 0 q q (iii) If G() 5, the fid the G( ) G() a (ii) valuate :, a 0 a a ( c ) it whe : (i) Divide b greatest power of i umerator ad deomiator. (ii) Put / ad appl 0 Illustratio 7 : valuate : Solutio : 5 5, form Put it 0 5 Illustratio 8 : If ( a b) F HG I KJ, the (A) a, b (B) a, b (C) a, b (D) oe of these Solutio : ( a b) F HG I KJ ( a) b a ( b) A s. ( ) a ( b) a b a 0, b a, b As. (C) Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65 Do ourself - 4 : (i) valuate : (ii) valuate : ( d ) Squeeze pla theorem (Sadwich theorem) : a b g b g b g b g h a b g the a si 0 as ( ) 0 0 0 0 5 ( ) Statemet : If f g h ; i the eighbourhood at a ad f gbg,. si 0, 0 si si lies betwee & 0 si
J-Mathematics. si 0 0 si lies betwee & si si si 0 0 as ( ) 0 0 0 [] [] []...[] Illustratio 9 : valuate : Where [.] deotes the greatest iteger fuctio. Solutio : We kow that < [] + +... < r [r] +...+ ( + ) < [r] r.( ) < [r] r Now, ad [] []... [] Thus, A s. 7. LIMIT OF TRIGONOMTRIC FUNCTIONS : si ta ta si [where is measured i radias] 0 0 0 0 ( a ) If f() 0, the a cot Illustratio 0 : valuate : 0 cos Solutio : cos 0 si cos si f(), e.g. a f() ( ) si( ) si Illustratio : valuate : 0 Solutio : (si( ) si ) si( ) 0 si( ) ( ) cos cos 0.cos( + cos ) A s. si.si 0 si.. cos si si( ) 0 cos si si( ) cos + si A s. 0 0 6 Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65
J-Mathematics Illustratio : valuate : a si b ta Solutio : As, 0 ad a also teds to zero si a should be writte as The give it F I HG K J Illustratio : cos si 4 4 a si a a si a a si a F H G I K J is equal to - so that it looks like 0 b b. ta b b. a ta b a( ).b si a b a b (A) (B) (C) (D) oe of these F I HG K J F H G I K J F I si HG K J Solutio : si cos si 4 4 si, where 4 4 0 4 A s. As. (B) Do ourself - 5 : (i) valuate : ( a ) si ta 0 ( b ) si si ( c ) h0 (a h) si(a h) a si a h Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65 ( b ) Usig substitutio f() f(a h) or f(a h) i.e. b substitutig b a h or a + h a h0 h0 Illustratio 4 : valuate : Solutio : (sec ta ) (sec ta );( form) si 0 ; ow i form cos 0 Put h si h it cosh h0 h 0 cos h sih 7
J-Mathematics h h si si h0 h h 0 h0 si cos h cos A s. 8. LIMIT USING SRIS XPANSION : pasio of fuctio like biomial epasio, epoetial & logarithmic epasio, epasio of si, cos, ta should be remembered b heart which are give below : a a a ( a ) a... a 0 ( b ) e!!!...!!! 4 b g... ( d ) si ( c ) for 4 (e) 4 6 cos...! 4! 6! 5 7 ( g ) ta... 5 7 (h ) si (i) sec. 5.. 5 7...! 5! 7! 4 6 5 6...! 4! 6! ( j ) ( +) + + ( )! +... Q 5 7...! 5! 7! 5 ( f ) ta... 5 e e Illustratio 5 : 0 si Solutio : e e 0 si 5..... 6 5! 0 5... 6 5!......!!!! 0 5...! 5!... 60 / 0 / 6... 6 0 Do ourself - 6 : si (i) valuate : 0 si( (ii) valuate : ) 0 9. LIMIT OF XPONNTIAL FUNCTIONS : a e ( a ) a(a 0) I particular. 0 0 I geeral if f() 0,the a f( ) a a f() a, a 0 8 ta A s. Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65
J-Mathematics ta e e Illustratio 6 : valuate : 0 ta ta Solutio : e e 0 ta (ta ) e e e ta 0 ta e (e ) ta 0 0 0 e (e ) e 0 [as 0, ta 0] where ta ad 0 e A s. Do ourself - 7 : (i) valuate : a e e a a (ii) valuate : 0 ( ) / ( b ) (i) / 0 e (Note : The base ad epoet depeds o the same variable.) I geeral, if f() 0, the a a / f() ( f()) e (ii) ( ) 0 (iii) If f() ad (), the ; () k f() e a where a a k () [f() ] a Illustratio 7 : valuate (log ) log Solutio : (log ) log (log log ) ( log ) log / log e b log a A s. log b a Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65 Illustratio 8 : valuate : Solutio : Illustratio 9 : valuate : Solutio : ( ta ) 0 cos ( ta ) 0 cos ( ta ) ta. cos ta 0. 4 Sice it is i the form of 4 e 4 A s. (4 + ) e 8 A s. 9
J-Mathematics si a Illustratio 0 :,a, a si a is a iteger, equals - Solutio : (A) cot a e (B) ta a e a a si si si a a si a a si a (C) si a e si si a a si a (D) cos a e si si a si a a si a si si a Illustratio : a a cos si. a e a si a 0 F HG a b c I KJ / e cos a si a e cot a As. (A) Solutio : (A) abc (B) abc (C) (abc) / (D) oe of these 0 F HG e 0 / a b c L F HG NM I KJ / 0 F HG a b c I KJ / a b c ( a ) ( b ) ( c ) ( ) ( ) ( ) L NM O QP I KJ O QP a b c a b c 0 e / (log a + log b + log c) e log (abc)/ (abc) / As. (C) Do ourself - 8 : (i) valuate : { ( a) } (ii) valuate : ta 4 0 / (iv) valuate : ( c ) If a 0 Illustratio : valuate : f() & 5 () ( ) 7 (f()) 0 5 0 (iii) valuate : ta ( v ) valuate : f() A 0 & () B a p q 4 6 (a fiite quatit) the ; f() 5 7 5 5 7 Solutio : Here f() 5,. () 7 a ( ) e B l A A B A s. Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65
J-Mathematics Do ourself - 9 : (i) valuate : 5 Miscellaeous Illustratios : 0 Illustratio : Fid... 4 r, where r +, 0 r ( ), r I, N Solutio : Let 0 cos the its 0 cos cos,... cos si, cos cos cos...cos 4 8 si si si. si cos 0 A s. Illustratio 4 : valuate : cos { cos ( cos (...( cos ())))} 0 4 si Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65 Solutio : Let A cos { cos ( cos (...( cos ())))} 0 4 si cos {si (si (...( cos ())))} 4 4 si. 4 0 Illustratio 5 : valuate the followig its, if eist Solutio : cos 0 0 As. si 4 4 ( 4 4) si ( )... ( )... ( ).... e. e. 4 e......... 4..... e e A s.
J-Mathematics Illustratio 6 : Solutio : valuate si. 0 Agai the fuctio f() si(/) is udefied at 0. valuatig the fuctio for some small values of, we get f() si 0, f si 0, f(0.) si0 0, f(0.0) si00 0. O the basis of this iformatio we might be tempted to guess that si 0 but this time 0 our guess is wrog. Note that although f(/) si 0 for a iteger, it is also true that f() for ifiitel ma values of that approach 0. [I fact, si(/) whe ad solvig for, we get /(4 + )]. The graph of f is give i followig figure si( /) The dashed lie idicate that the values of si(/) oscillate betwee ad ifiitel ofte as approaches 0. Sice the values of f() do ot approach a fied umber as approaches 0, si does ot eist. 0 ANSWRS FOR DO YOURSLF : ( i ) (a) T (b) F (c) F (d) T (e) T (f) T (g) T (h) T (i) F (j) T : (i) : (i) q p (ii) 4 : (i) (ii) 5 : (i) ( a ) 6. (i) (ii) 6 7 : (i) e a (ii) ( b ) si (iii) 8 : (i) a (ii) e (iii) e p (iv) 9 : (i) 0 4 ( c ) asia + a cosa e ( v ) e 5 Node-6\:\Data\04\Kota\J-Advaced\SMP\Maths\Uit#04\g\0.LIMIT\0.THORY.p65