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Gunakan teorema apit!<br />
c<br />
c<br />
2. lim cos x = 1<br />
( 4.16 )<br />
x→ 0<br />
3. lim sin x = 0<br />
( 4.17 )<br />
x→ 0<br />
4. lim tan x = 0<br />
x→0<br />
Bukti : lim tan x = lim<br />
0 x→0<br />
tan x<br />
5. lim = 1<br />
x→0 x<br />
Bukti :<br />
tan x<br />
=<br />
x<br />
lim<br />
x→0<br />
sin x<br />
cos<br />
=<br />
x→ x<br />
lim sin x<br />
x→0<br />
lim 1<br />
x→0<br />
= lim sin x . =<br />
x→ 0 lim cos x<br />
x→0<br />
sin x<br />
lim<br />
x→0<br />
x<br />
1<br />
.<br />
cos x<br />
=<br />
sin x<br />
lim<br />
x→0 x<br />
. lim<br />
x→0<br />
1<br />
cos x<br />
1<br />
(0) = 0<br />
1<br />
(terbukti)<br />
1<br />
. lim = 1 . 1 = 1 (terbukti)<br />
x→0 cos x<br />
( 4.18 )<br />
( 4.19 )<br />
x<br />
6. lim = 1<br />
x→0 tan x<br />
Bukti :<br />
x<br />
=<br />
tan x<br />
lim<br />
x→0<br />
1<br />
lim<br />
x→0<br />
sin x<br />
x<br />
.cosx = 1 . 1 = 1 (terbukti)<br />
( 4.20 )<br />
cos x -1<br />
7. lim = 0<br />
( 4.21 )<br />
x→ 0 x<br />
Bukti :<br />
2 1 2 1<br />
cos x - sin x -1<br />
cos x -1<br />
lim = lim<br />
2 2<br />
=<br />
x→0 x x→ 0<br />
x<br />
2 1<br />
1 1<br />
1<br />
-2sin<br />
x -2sin<br />
x . sin x<br />
sin x<br />
1<br />
lim<br />
2<br />
= lim<br />
2 2<br />
= lim -sin x<br />
2<br />
= 0(1)= 0<br />
x→0<br />
x<br />
x→0<br />
1<br />
x→0<br />
2 1<br />
2( x)<br />
x<br />
2<br />
2<br />
(terbukti)<br />
85
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