-
第2ç« æ¥µé
äžè§é¢æ°ã𿥵é
1 颿°ã®æ¥µéãšå€§å°é¢ä¿
limf(x) =α, limg(x) =β ãšããã
xa
pix
1
xãαã«è¿ããšã,åžžã« f(x) âŠg(x)ãªãã° aâŠÎ²
2xãαã«è¿ããšã,åžžã« f(x) (x)g(x) ãã€Î±=β ãªãã° limh(x)=a
泚æ äžã®äºæã¯,xââ, xââã®å Žåã«ãæãç«ã€ã
â æ¬¡ã®æ¥µéãæ±ããã [104, 105]
1-cos 3x
â¡ 104(1) lim
xâ0
x2
1
*105(1) limxcos
0+x
x
第2ç¯ é¢æ°ã®æ¥µé
31 0
(2) lim
sinx2
x01âcosx
(2) lim
1+sinx
XIIâ
x
第2ç«
極é
泚æ2ããã¯ãã¿ãã¡ã®åçã ãšããããšãããã
äŸé¡
3 limf(x)=â ã®ãšã,åå倧ããxã§åžžã« f(x)âŠg(x) ãªãã° limg(x) =â
|2 äžè§é¢æ°ã𿥵é
sinx
lim
x0 x
x
=1, lim -1 (è§ã®åäœã¯ã©ãžã¢ã³)
x-0 sinx
STEPA
äžå¿ã 0, çŽåŸ ABã4ã®ååã®åŒ§ã®äžç¹ãMãšã, Aããåºãå
ç·
ã匧 MB äžã®ç¹Pã§åå°ããŠ, ABäžã®ç¹Qã«ãããšããã
(1) 0=â PAB ãšãããšã, OQ ã®é·ãã0ã§è¡šãã
(2) PBã«éããªãè¿ã¥ããšã, Qã¯ã©ããªç¹ã«è¿ã¥ããŠãããã
|æé Aããåºãå
ç·ã MBäžã®ç¹Pã§åå°ããŠ, ABäžã®ç¹Qã«ãããšã
â OPA = â OPQ
sin O
æ±ãããã®ãåŒã§è¡šãã
ãªã©ã®æ¥µéã«åž°çãããã
è§£ç (1) å³ã®å³ã«ãããŠ
â 99 æ¬¡ã®æ¥µéã調ã¹ãã
ZOQ= â OPA=â OAP=0
â PQB= â PAQ+ â APQ=30
M
2
(1) lim cos-
*(2) lim
(3)lim
x
tanx
xâ0 sinx
ãã£ãŠ â OQP=30
â³OPQã«æ£åŒŠå®çãçšãããš,P=2 ã§ãããã
30
0 Q B
â æ¬¡ã®æ¥µéãæ±ããã [ 100~103]
â 100 (1) lim
xâ0
sin 4x
XC
sin2x
*(2) lim
x-0 sin5x
(3) lim
x-0 tant
sin3x
tan2x-sinx
â¡ 101 (1) lim-
*(2) lim
xâ0
x
1-cos 2x
x-0 xsinx
(3) lim
xâ0
sin3x+sinx
sin2x
â¡ 102(1) lim
COS X
x-Sin2x
(2) lim-
sin2x
(3) lim
x01âcosx
103*(1) lim
tan x
X10 x
*(4) lim-
sinлx
x-1 x-1
1âcosx
t- sinx
STEPB
*(2) lim
XâÏ
OQ
2
sin O sin(-30)
ãŸã, sin (Ï-30)=sin30 ã§ãããã
2sin
OQ=
sin 30
(2)PãBã«éããªãè¿ã¥ããšã, 0 +0 ã§ããã ãã®ãšã
2 sin
2 sin
3 2
lim OQ= lim
lim
8+0
o sin 30
0-40 3 0 sin 36 3
ãã£ãŠ,Qã¯ç·å OBäžã®0ããã®è·é¢ã«ããç¹ã«è¿ã¥ããŠãããå
â¡ 106 ååŸÎ±ã®ååšäžã«åç¹Pãšå®ç¹Aãããã Aã«ãããæ¥ç·äžã«
AQ=AP ã§ãããããªç¹QãçŽç·OAã«é¢ããŠPãšåãåŽã«ãšããPãA
PQ
ã«éããªãè¿ã¥ããšã,
AP
ã®æ¥µéå€ãæ±ããã ãã ã,P㯠â AOP
(0<< AOP < 1)ã«å¯Ÿãã匧AP ã®é·ãã衚ãã
sin(x-7)
x-Ï
(3) lim x-- tanx
xn
ax+b
1
sin(sinx)
(5) lim
xâ0 sinx
1
107 çåŒ lim
(6) limxsin
COS x
2x
ãæãç«ã€ããã«, 宿°a, b ã®å€ãå®ããã