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FBGBKL?JKL?J:PBB :JKL
F:L?F:LBQ?KDBCN:DMEVL?L D:N?>J:MJ:GUOB L?HJBB
ih\ukr_cfZl_fZlbd_^eyklm^_glh\dmjkZ aZhqgh]hhl^_e_gby]_heh]bq_kdh]hnZdmevl_lZ QZklv
KhklZ\bl_eb =;KZ\q_gdh K:LdZq_\Z
-2<\_^_gb_ GZklhysb_ f_lh^bq_kdb_ mdZaZgby ij_^gZagZq_gu ^ey klm^_glh\-aZhqgbdh\ ]_heh]bq_kdh]h nZdmevl_lZ b kh^_j`Zl h[sb_ f_lh^bq_kdb_ mdZaZgby ih bamq_gbx jZa^_eh\ \ukr_c fZl_fZlbdb ©:gZeblbq_kdZy ]_hf_ljbyª ©
(x2 − x1 )2 + ( y1 − y 2 )2
= M 1M 2 .
(1)
>_e_gb_hlj_adZ\^Zgghfhlghr_gbb >Zgulhqdb M 1 (x1 ; y1 ) b M 2 (x 2 ; y 2 ) =h\hjylqlhlj_lvylhqdZ M (x; y ) e_`ZsZy gZ ^Zgghc ijyfhc ^_ebl hlj_ahd M 1 M 2 \ M M ( λ -iheh`bl_evgh_keblhqdZ hlghr_gbb λ _keb λ = ± 1 MM 2 M e_`blf_`^mlhqdZfb M 1 b M 2 bhljbpZl_evgh_keblhqdZ M e_`bl \g_ hlj_adZ M 1 M 2 Dhhj^bgZlu lhqdb M (x; y ) , ^_eys_c hlj_ahd M 1 M 2 \ hlghr_gbb λ hij_^_eyxlky ih nhjfmeZf x + λx 2 y + λy 2 (2) x= 1 ; y= 1 , (λ ≠ −1) . 1+ λ 1+ λ Dhhj^bgZluk_j_^bguhlj_adZhij_^_eyxlkyihnhjfmeZf x + x2 y + y2 ;y = 1 x= 1 . (3) 2 2 IehsZ^v lj_m]hevgbdZ k \_jrbgZfb A(x1 ; y1 ), B (x 2 ; y 2 ), C (x3 ; y 3 )gZoh^blkyihnhjfme_ 1 S = x1 ( y 2 − y3 ) + x 2 ( y3 − y1 ) + x3 ( y1 − y 2 ) . (4) 2
H[s__mjZ\g_gb_ijyfhc MjZ\g_gb_\b^Z Ax + By + C = 0,
-3-
(5)
]^_ A, B, C -ihklhyggu_dhwnnbpb_glu A 2 + B 2 ≠ 0 ; x b y dhhj^bgZlu ex[hc lhqdb hij_^_ey_l gZ iehkdhklb g_dhlhjmx ijyfmx MjZ\g_gb_ gZau\Z_lkyh[sbfmjZ\g_gb_fijyfhc. MjZ\g_gb_ijyfhckm]eh\ufdhwnnbpb_glhf MjZ\g_gb_\b^Z y = kx + b (6) gZau\Z_lky mjZ\g_gb_f ijyfhc k m]eh\uf dhwnnbpb_glhf. k – m]eh\hc dhwnnbpb_gl k = tgα ( α - m]he f_`^m ijyfhc b iheh`bl_evgufgZijZ\e_gb_fhkbOx ). M]hef_`^mijyfufb Hkljucm]he ϕ f_`^mijyfufb y = k1 x + b1 b y = k 2 x + b2 hij_^_ey_lkyihnhjfme_ k −k (7) tgϕ = 2 1 , k1k 2 ≠ −1 . 1 + k1k 2 Mkeh\b_iZjZee_evghklbijyfuo (8) k1 = k 2 . Mkeh\b_i_ji_g^bdmeyjghklbijyfuo 1 k1 = − beb k1k 2 = −1 . (9) k2 MjZ\g_gb_ijyfhckm]eh\ufdhwnnbpb_glhf k ijhoh^ys_c q_j_a^Zggmxlhqdm M 0 (x0 ; y 0 )bf__l\b^ ( 10 ) y − y 0 = k (x − x0 ), k ≠ 0 . MjZ\g_gb_ijyfhcijhoh^ys_cq_j_a^\_lhqdb MjZ\g_gb_ ijyfhc ijhoh^ys_c q_j_a ^\_ aZ^Zggu_ lhqdb M 1 (x1 ; y1 ) b M 2 (x 2 ; y 2 ) bf__l\b^ y − y1 x − x1 ( 11 ) ; x2 ≠ x1 ; y 2 ≠ y1 . = y 2 − y1 x2 − x1 M]eh\hcdhwnnbpb_glwlhcijyfhchij_^_ey_lkyihnhjfme_ y − y1 k= 2 . x2 − x1
-4MjZ\g_gb_ijyfhc\hlj_adZo MjZ\g_gb_\b^Z x y ( 12 ) + = 1; a ≠ 0; b ≠ 0 a b gZau\Z_lkymjZ\g_gb_fijyfhc\hlj_adZo A^_kv a b b -Z[kpbkkZbhj^bgZlZlhqdbi_j_k_q_gbyijyfhc khkvxOx bhkvxOy khhl\_lkl\_ggh GhjfZevgh_mjZ\g_gb_ijyfhc MjZ\g_gb_\b^Z ( 13 ) x cosα + y sin α − p = 0 gZau\Z_lky ghjfZevguf mjZ\g_gb_f ijyfhc A^_kv p - ^ebgZ i_ji_g^bdmeyjZhims_ggh]hbagZqZeZdhhj^bgZlgZijyfmx α m]hef_`^mwlbfi_ji_g^bdmeyjhfbiheh`bl_evgufgZijZ\e_gb_f hkbOx . Qlh[u h[s__ mjZ\g_gb_ ijyfhc ijb\_klb d ghjfZevghfm \b^m gZ^h\k_qe_gumjZ\g_gby mfgh`blvgZghjfbjmxsbc 1 . fgh`bl_ev M = ± A2 + B 2 >ey M gZ^h\aylv©ª_keb C < 0 agZd©-ª_keb C > 0 . JZkklhygb_hllhqdb^hijyfhc JZkklhygb_ d hl ^Zgghc lhqdb M 0 (x0 , y 0 ) ^h ijyfhc Ax + By + C = 0 hij_^_ey_lkyihnhjfme_ Ax0 + By 0 + C d= . ( 14 ) 2 2 A +B Hdjm`ghklv Hdjm`ghklv – wlh fgh`_kl\h lhq_d iehkdhklb M (x; y ) , jZ\ghm^Ze_gguohl^Zgghclhqdb C (a; b ) . MjZ\g_gb_hdjm`ghklbbf__l\b^ ( 15 ) (x − a )2 + ( y − b )2 = R 2 , C (a; b ) -p_gljhdjm`ghklb R -jZ^bmkhdjm`ghklb Weebik Weebik – wlh fgh`_kl\h lhq_d iehkdhklb M (x; y ) kmffZ jZkklhygbc dhlhjuo ^h ^\mo lhq_d F1 (− c;0 ) b F2 (c;0 ) _klv \_ebqbgZihklhyggZyjZ\gZy 2a (2a > 2c ) .
-5DZghgbq_kdh_ ijhkl_cr__ mjZ\g_gb_weebikZbf__l\b^ x2 y2 ( 16 ) + = 1. a 2 b2 A^_kv a, b - ihemhkb weebikZ F1 b F2 - nhdmku weebikZ c = ε < 1 lZd dZd a > c gZau\Z_lky a 2 = c 2 + b 2 Qbkeh a wdkp_gljbkbl_lhfweebikZ NhdZevgu_jZ^bmku r1 = F1 M b r2 = F2 M hij_^_eyxlkyih nhjfmeZf r1 = a + εx; r2 = a − εx. =bi_j[heZ =bi_j[heZ – wlh fgh`_kl\h lhq_d iehkdhklb M (x; y ) , Z[khexlgZy \_ebqbgZ jZaghklb jZkklhygbc dhlhjuo ^h ^\mo lhq_d F1 (− c;0 ) b F2 (c;0 ) _klv \_ebqbgZ ihklhyggZy jZ\gZy 2a ( 2 a < 2c ) . F1 M − F2 M = 2a . F1 b F2 -nhdmku]bi_j[heu r1 = F1 M b r2 = F2 M -nhdZevgu_ jZ^bmku DZghgbq_kdh_mjZ\g_gb_]bi_j[heu x2 y2 − = 1. ( 17 ) a2 b2 A^_kv a, b - ihemhkb ]bi_j[heu ^_ckl\bl_evgZy b fgbfZy khhl\_lkl\_ggh c2 = a2 + b2 . c Qbkeh = ε > 1 lZddZd a < c ) –wdkp_gljbkbl_l]bi_j[heu a NhdZevgu_jZ^bmkuhij_^_eyxlkyihnhjfmeZf r1 = εx + a ; r2 = εx − a . =bi_j[heZ khklhbl ba ^\mo \_l\_c jZkiheh`_gguo hlghkbl_evgh hk_c dhhj^bgZl LhqdZ O - p_glj ]bi_j[heu Lhqdb A1 (− a;0 ) b A2 (a;0 ) - \_jrbgu i_j_k_q_gby k hkvx Ox b ]bi_j[heu=bi_j[heZbf__l^\_Zkbfilhlu y = ± x ?keb a = b , a lh]bi_j[heZgZau\Z_lkyjZ\ghklhjhgg_c =bi_j[heu
-62
2
2
2
x y y x 1 − = − =1 b a2 b2 b2 a2 gZau\Zxlkykhijy`_ggufb IZjZ[heZ IZjZ[heZ – wlh fgh`_kl\h lhq_d iehkdhklb M (x; y ) , p jZ\ghm^Ze_gguohl^Zgghclhqdb F ,0 gZau\Z_fhcnhdmkhfb 2 p ^Zgghcijyfhc x = − gZau\Z_fhc^bj_dljbkhc 2 DZghgbq_kdh_mjZ\g_gb_iZjZ[heubf__l\wlhfkemqZ_\b^ y 2 = 2 px , FM = r -nhdZevgucjZ^bmkhij_^_ey_lkyihnhjfme_ p r = x + , ( p > 0 ). 2
\lhjh]h ihjy^dZ gZau\Z_lky qbkeh a a12 h[hagZqZ_fh_kbf\hehf 11 bhij_^_ey_fh_jZ\_gkl\hf a 21 a 22 a11 a12 = a11a 22 − a 21a12 . ( 1) a 21 a 22 Hij_^_ebl_e_f lj_lv_]h ihjy^dZ gZau\Z_lky qbkeh hij_^_ey_fh_jZ\_gkl\hf a11 a12 a13 a a a a a a a21 a22 a23 = a11 22 23 − a12 21 23 + a13 21 22 .( 2 ) a32 a33 a31 a33 a31 a32 a31 a32 a33 Kbkl_fu^\moebg_cguomjZ\g_gbck^\mfyg_ba\_klgufb Kbkl_fZ a11 x + a12y = b 1 (3) a x a y b + = 21 22 2 bf__lj_r_gb_
b1 a12 b a 22 , x= 2 a11 a12 a 21 a 22 ]^_ ∆ =
a11 a 21
a11 a y = 21 a11 a 21
-7b1 b2 , a12 a 22
(4)
a12 . a 22
Kbkl_fZ ^\mo h^ghjh^guo ebg_cguo mjZ\g_gbc k lj_fy g_ba\_klgufb a11 x + a12 y + a13 z = 0 (5) 0 a x a y a z + + = 22 23 21 bf__lj_r_gb_ a a13 a a13 a a12 , y = −t 11 , z = t 11 , x = t 12 a 22 a 23 a21 a23 a 21 a 22 ]^_ t -ijhba\hevgh_qbkeh Kbkl_fZ lj_o h^ghjh^guo ebg_cguo mjZ\g_gbc k lj_fy g_ba\_klgufb a11 x + a12 y + a13 z = 0 a 21 x + a 22 y + a 23 z = 0 a x + a y + a z = 0 32 33 31 bf__l hlebqgu_ hl gmey j_r_gby lh]^Z b lhevdh lh]^Z dh]^Z hij_^_ebl_evkbkl_fu a11 a12 a13 (7) ∆ = a 21 a 22 a 23 = 0. a31 a32 a33 Kbkl_fZlj_oebg_cguomjZ\g_gbck^\mfyg_ba\_klgufb a11 x + a12 y = b1 (8) a 21 x + a 22 y = b2 a x + a y = b 32 3 31 a11 a12 b1 kh\f_klgZ dh]^Z a 21 a 22 b2 = 0 b kbkl_fZ g_ kh^_j`bl a31 a32 b3 ihiZjghijhlb\hj_qb\uomjZ\g_gbc
-8Kbkl_fZlj_omjZ\g_gbcklj_fyg_ba\_klgufb a11 x + a12 y + a13 z = b1 a 21 x + a 22 y + a 23 z = b2 a x + a y + a z = b 32 33 3 31 ijbmkeh\bbqlh a11 a12 a13 ∆ = a 21 a 22 a 23 ≠ 0 a31 a32 a33 bf__l_^bgkl\_ggh_j_r_gb_ ∆y ∆ ∆ x= x, y= , z= z , ∆ ∆ ∆ ]^_ b1 a12 a13 a11 b1 a13 a11 ∆ x = b2 a 22 a 23 , ∆ y = a 21 b2 a 23 , ∆ z = a 21 b3 a32 a33 a31 b3 a33 a31
(9)
( 10 ) a12 a 22 a32
b1 b2 . b3
G_kh\f_klgu_bg_hij_^_e_ggu_kbkl_fu Imklv hij_^_ebl_ev kbkl_fu ∆ = 0 Lh]^Z \hafh`gu ke_^mxsb_kemqZb we_f_glu ^\mo kljhd hij_^_ebl_ey ijhihjpbhgZevgu a a a gZijbf_j 11 = 12 = 13 = m Lh]^Z a 21 a 22 a 23 Z _keb b1 ≠ mb2 lhkbkl_fZg_kh\f_klgZ [ _keb b1 = mb2 lhkbkl_fZg_hij_^_e_ggZ _kebi_j\h_b lj_lv_mjZ\g_gbyg_ijhlb\hj_qb\u < hij_^_ebl_e_ ∆ g_l kljhd k ijhihjpbhgZevgufb we_f_glZfbLh]^Zkms_kl\mxlqbkeZ C1 b C 2 hlebqgu_hlgmey ijbdhlhjuo mL1 + nL2 = L3 b Z _keb mb1 + nb2 ≠ b3 lhkbkl_fZg_kh\f_klgZ [ _keb mb1 + nb2 = b3 lh kbkl_fZ g_hij_^_e_ggZ ]^_ Li (i = 1,2,3) -e_\u_qZklbmjZ\g_gby Dhfie_dkgu_qbkeZ Hij_^_e_gb_ Dhfie_dkguf qbkehf gZau\Zxl qbkeZ \b^Z a + ib ]^_ a b b - ^_ckl\bl_evgu_ qbkeZ Z i 2 = −1 fgbfZy _^bgbpZ ( 11 ) i 3 = −i; i 4 = 1; i 5 = i bl^
-9Keh`_gb_ \uqblZgb_ mfgh`_gb_ b \ha\_^_gb_ \ kl_i_gv dhfie_dkguo qbk_e \uihegyxl ih ijZ\beZf wlbo ^_ckl\bc gZ^ fgh]hqe_gZfbkaZf_ghckl_i_g_cqbkeZ i ihnhjfmeZf Ljb]hghf_ljbq_kdZynhjfZdhfie_dkgh]hqbkeZ Dhfie_dkgh_ qbkeh z = a + ib hij_^_ey_lky iZjhc \_s_kl\_gguoqbk_e (a; b )bihwlhfmbah[jZ`Z_lkylhqdhc M (a; b ) iehkdhklbbeb__jZ^bmkhf\_dlhjhf r = OM >ebgZwlh]h\_dlhjZ gZau\Z_lkyfh^me_fdhfie_dkgh]hqbkeZ r = a 2 + b 2 Zm]he ϕ k hkvx Ox gZau\Z_lky Zj]mf_glhf dhfie_dkgh]h qbkeZ LZd dZd x = r cos ϕ , y = r sin ϕ lh ( 12 ) z = r (cos ϕ + i sin ϕ ). >_ckl\by gZ^ dhfie_dkgufb qbkeZfb \ ljb]hghf_ljbq_kdhc nhjf_ r1 (cos ϕ1 + i sin ϕ1 )⋅ r2 (cos ϕ 2 + i sin ϕ 2 ) = 1) , ( 13 ) r1 ⋅ r2 [cos(ϕ1 + ϕ 2 ) + i sin (ϕ1 + ϕ 2 )] 2)
r1 (cos ϕ1 + i sin ϕ1 ) r1 = [cos(ϕ1 − ϕ 2 ) + i sin (ϕ1 − ϕ 2 )] , ( 14 ) r2 (cos ϕ 2 + i sin ϕ 2 ) r2
[r (cos ϕ + i sin ϕ )]n
= r n (cos nϕ + i sin nϕ ) , ϕ + 2kπ ϕ + 2kπ 4) Wk = n r (cos ϕ + i sin ϕ ) = n r cos + i sin n n ]^_ k = 0,1,2,..., (n − 1).
3)
( 15 ) ,( 16 )
3. FZl_fZlbq_kdbcZgZeba Ij_^_eu Qbkeh A gZau\Z_lky ij_^_ehf nmgdpbb f (x) ijb x → a , _keb^eyex[h]hkdhevm]h^ghfZeh]h ε > 0 gZc^_lkylZdh_ δ > 0 , qlh ijb 0 < x − a < δ ⇒ f ( x) − A < ε Ibrml lim f ( x) = A . x→a
IjZdlbq_kdh_ \uqbke_gb_ ij_^_eh\ hkgh\u\Z_lky gZ ke_^mxsbo l_hj_fZo ?kebkms_kl\mxldhg_qgu_ij_^_eu lim f ( x) b lim g ( x) lh x→a
1)
lim[ f ( x ) + g ( x)] = lim f ( x) + lim g ( x) , x→a
x →a
x→a
x→a
(1)
2)
- 10 lim[ f ( x ) ⋅ g ( x)] = lim f ( x ) ⋅ lim g ( x ) ,
(2)
3)
f ( x) f ( x) lim x→a = lim ijb lim g ( x) ≠ 0 ) . x→a g ( x) x→a lim g ( x)
(3)
x→a
x→a
x→a
x →a
BkihevamylZd`_ke_^mxsb_ij_^_eu ln(1 + α ) sin α 1) lim = 1, 3) lim =1 , α →0 α α →0 α 2)
1 α
lim (1 + α ) = e ,
α →0
5)
4)
aα − 1 lim = ln a α →0 α
m 1+ α ) −1 ( lim =m
α A^_kv α = α (x) -[_kdhg_qghfZeZynmgdpby lim α (x ) = 0 .
(
x →a
(4)
α →0
.
)
KjZ\g_gb_[_kdhg_qghfZeuo Imklv α (x) b β (x) [_kdhg_qghfZeu_ijb x → a ?keb α ( x) = 1 lh [_kdhg_qgh fZeu_ gZau\Zxlky wd\b\Ze_glgufb lim x→a β ( x) Ibrml α ~ β . L_hj_fZ ?keb hlghr_gb_ ^\mo [_kdhg_qgh fZeuo bf__l ij_^_e lh wlhl ij_^_e g_ baf_gblky ijb aZf_g_ dZ`^hc ba [_kdhg_qghfZeuowd\b\Ze_glghc_c[_kdhg_qghfZehclh_klv_keb α lim = m,α ~ α1 , β ~ β1 lh x →a β α α lim 1 = lim = m . (5) x →a β x →a β 1 Ihe_agh bkihevah\Zlv wd\b\Ze_glghklv ke_^mxsbo [_kdhg_qghfZeuo _keb α → 0 lh sin α ~ α , tgα ~ α , arcsin α ~ α , arctgα ~ α , ln(1 + α ) ~ α , aα − 1 ~ α ln a,
(1 + α )m − 1 ~ α ⋅ m.
( 5/ )
>bnn_j_gpbjh\Zgb_nmgdpbc Hij_^_e_gb_Ijhba\h^ghchlnmgdpbb y = f (x) \lhqd_ x gZau\Z_lkydhg_qgucij_^_e ∆y f (x + ∆x ) − f ( x) lim (6) = lim = f / ( x) . ∆x →0 ∆x →0 ∆x ∆x
- 11 GZoh`^_gb_ ijhba\h^ghc gZau\Z_lky ^bnn_j_gpbjh\Zgb_f nmgdpbb Hkgh\gu_ijZ\beZ^bnn_j_gpbjh\Zgby Imklv C = const , u = u (x), v = v(x) - ^bnn_j_gpbjm_fu_ nmgdpbb Lh]^Z / / 3) (u ± v ) = u / ± v / ; 1) C / = 0; 2) (Cu ) = Cu / ; /
/ / u u v − uv 5) = ; 4) (uv ) = u v + uv ; v2 v _keb y = f (u ), u = u (x), lh y / ( x) = y / (u ) ⋅ u / ( x) ijZ\beh^bnn_j_gpbjh\Zgbykeh`ghcnmgdpbb /
/
/
(7)
Ijhba\h^gZykl_i_ggh-ihdZaZl_evghcnmgdpbb
(u ) = v ⋅ u v /
v −1
⋅ u / + u v ⋅ ln u ⋅ v / , ]^_ u = u ( x), v = v( x) -^bnn_j_gpbjm_fu_nmgdpbb
(8)
AZ^Zqbkj_r_gb_f AZ^ZqZGZclbm]hef_`^mijyfufb : 4 x + 2 y − 5 = 0 b 6 x + 3 y + 1 = 0 ; 3x − y − 2 = 0 b 3x + y − 1 = 0 . J_r_gb_ Ijb\_^_f mjZ\g_gby d \b^m mjZ\g_gbc ijyfuo k m]eh\ufdhwnnbpb_glhf i 5 : 4 x + 2 y − 5 = 0 ⇒ y = −2 x + ⇒ k1 = −2 2 1 6 x + 3 y + 1 = 0 ⇒ y = −2 x − ⇒ k 2 = −2 . 3 M]eh\u_ dhwnnbpb_glu wlbo ijyfuo jZ\gu ke_^h\Zl_evgh, ijyfu_iZjZee_evgu ϕ = 0. ; 3x + y − 2 = 0 ⇒ y = 3x − 2 ⇒ k1 = 3 3x + y − 1 = 0 ⇒ y = − 3x + 1 ⇒ k 2 = − 3 . Ihnhjfme_ i ihemqbf −2 3 − 3− 3 = = 3;ϕ = 60 o. tgϕ = −2 1− 3 ⋅ 3
- 12 AZ^ZqZ Wdkp_gljbkbl_l ]bi_j[heu jZ\_g 2 GZclb ijhkl_cr__mjZ\g_gb_]bi_j[heuijhoh^ys_cq_j_alhqdm M 2 ;1 .
(
)
c = 2 beb a c 2 = 2a 2 LZd dZd c 2 = a 2 + b 2 lh a 2 + b 2 = 2a 2 beb a 2 = b 2 . Ke_^h\Zl_evgh]bi_j[heZjZ\ghklhjhggyy Ih^klZ\bf dhhj^bgZlu lhqdb M 2 ;1 \ mjZ\g_gb_ i J_r_gb_ Wdkp_gljbkbl_l ]bi_j[heu
( )
(
)
2 − (1) = a 2 beb a 2 = 1 . ihemqbf 3) Bkdhfh_mjZ\g_gb_]bi_j[heubf__l\b^ x 2 − y 2 = 1 . 2
2
AZ^ZqZGZclb\k_agZq_gby 6 1 . J_r_gb_Ljb]hghf_ljbq_kdZynhjfZ z = 1 bf__l\b^ z = 1 cos 0 o + i sin 0 o
(
)
0 + 2kπ 0 + 2kπ + i sin Wk = 6 z = 6 1 cos = 6 6 kπ kπ 2kπ 2kπ = cos + i sin = cos + i sin , k = 0,1,2,...5 6 6 3 3 AZ^ZqZJ_rblvkbkl_fmmjZ\g_gbc 3 x + 2 y − z = 0 x + 2 y + 9 z = 0. x + y + 2z = 0 J_r_gb_Bf__f 3 2 −1 2 9 1 9 1 2 1 2 9 =3 −2 −1 = −15 + 14 + 1 = 0. 1 2 1 2 1 1 1 1 2 Ke_^h\Zl_evgh kbkl_fZ bf__l j_r_gby hlebqgu_ hl gme_\h]h J_rZ_f kbkl_fm i_j\uo ^\mo mjZ\g_gbc Lj_lv_ mjZ\g_gb_y\ey_lkyboke_^kl\b_f 3x + 2 y − z = 0 . x + 2 y + 9z = 0 IhnhjfmeZf i ihemqbf
- 13 2 −1 3 −1 3 2 x=t = 20t; y = −t = −28t; z = t = 4t. 2 9 1 9 1 2 sin (x − 3) . x →3 x 2 − 4 x + 3
AZ^ZqZGZclb lim
x − 3 → 0 ke_^h\Zl_evgh, J_r_gb_ Ijb x → 3 / sin( x − 3) ~ x − 3 i Bkihevamy i l_hj_fm h[ wd\b\Ze_glghklb[_kdhg_qghfZeuobf__f x−3 sin (x − 3) 1 1 = lim = lim = . lim 2 x →3 x − 4 x + 3 x →3 (x − 3)(x − 1) x →3 x − 1 2 cos 4 x − cos 2 x . x →0 arcsin 2 3x
AZ^ZqZGZclb lim
J_r_gb_Ihnhjfme_ljb]hghf_ljbb 4x + 2x 4x − 2x = −2 sin 3x sin x. cos 4 x − cos 2 x = −2 sin sin 2 2 sin 3 x ~ 3 x, sin x ~ x, arcsin 3x ~ 3x lh _klv Ijb x → 0 2 2 (arcsin 3x ) ~ (3x ) Ihwlhfm cos 4 x − cos 2 x 2 − 2 sin 3x sin x − 2 ⋅ 3x ⋅ x lim = lim = lim =− . 2 2 2 x →0 x →0 x →0 3 arcsin 3x arcsin 3x (3x ) AZ^ZqZGZclbijhba\h^gu_nmgdpbc
(
) y = ln (x + 5)
1) y = 2 x 3 + 5 2)
4
2
2
3) y = x x . J_r_gb_ H[hagZqbf 2 x 3 + 5 = u lh]^Z y = u 4 Ih ijZ\bem ^bnn_j_gpbjh\Zgbykeh`ghcnmgdpbb iZjZ]jZn bf__f
( ) ⋅ (2 x
y/ = u4
/ u
(ln(x
2
2)
+5
3
))
/
+5
=
)
/ x
( )
(
)
3
= 4u 3 6 x 2 = 24 x 2 2 x 3 + 5 .
2x . x2 + 5
- 14 A^_kvhkgh\Zgb_bihdZaZl_evaZ\bkylhl x ; u = x; v = x 2 . Ihnhjfme_ ihemqZ_f
(x ) = x (x ) = x x2
/
x2
/
2
⋅ xx
x 2 +1
2
−1
2
⋅ 1 + x x ⋅ ln x ⋅ 2 x
(1 + 2 ln x ) Dhgljhevgu_aZ^Zgby
(
)
x −1 x →1 x − 1 x −1 \ lim 3 x →1 x − 1 sin x − sin a ] lim x→a x−a x x ^ lim x →∞ x + 1 5. GZclbijhba\h^gu_nmgdpbc [ lim
Z y =
1 ln x x
3x − x 3 [ y = arctg 1 − 3x 2 1 1 \ y = ln1 − + x x ] y = ln sin xtg x − x ^ y = 2 xtg 2 x + ln cos 2 x − 2 x 2 .
1.
2. 3.
4.
- 15
(
[ lim x →1
3
)
x 2 − 23 x + 1
(x − 1)2
x2 − 2x + 6 − x2 + 2x − 6 \ lim x→3 x2 − 4x + 3 cos mx − cos nx ] lim x→0 x2 tgπx ^ lim x→−2 x + 2 GZclbijhba\h^gu_nmgdpbc Z y = cos 5 (7 x + 9) [ y = 3 x + x + 1 \ y = ln 3 (5 x + 1) x ] y = sin 3x + 9
^ y = (ctgx )
x3
x + 2y + z = 2 3. J_rblvkbkl_fmmjZ\g_gbc 3x + 2 y + z = 4 4 x + 3 y − 2 z = 9
- 16 -
4. GZclb x
x +8 Z lim x →∞ x − 2 sin x − cos x [ lim π π − 4x x→ 4
4 + x + x2 − 2 \ lim x→−1 x +1 1 − cos 5 x ] lim x→0 1 − cos 3 x sin 2 x ^ lim x→0 ln(1 + x) 5. GZclbijhba\h^gu_nmgdpbc Z y = arccos 1 − 2 x [ y = cos 5 (3 x + 1) \ y = x arcsin x x +1 ] y = x+ x ^ y = 2 cos
2
5
x −3 cos x
.
Ebl_jZlmjZ 1. RbiZq_\ <K
