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&!" +"*& 0%2 .'7!& +'6'&) "M')& 0%2 0." 1$&! "O(0+ &$ I )$
lim f (x) = 0 = f (2).
x→2
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<0+(" 0& S3 T$
lim g(x) = 6 6= g(2).
x→2
4% &!" &!'.2 0%2 *$(.&! "M06;+": &!" +"*& 0%2 .'7!& +'6'&) 0." 2'U"."%&: )$ x→0
lim h(x) 0%2
lim k(x) 2$ %$& "M')&3
x→0
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&$ &!" .'7!& )'2"3 V$. &!" *(%#&'$% k(x): 0) -" 0;;.$0#! 0 *.$6 &!" +"*&: &!" *(%#&'$% 7$")
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)*+"$#$!" !' ,!"#$"%$#-
X"& f 1" 0 ."0+9<0+("2 *(%#&'$% 0%2 a 1" 0 ."0+ %(61".3 W" )0, f (x) ')
lim f (x) = f (a)3
x→a
!
!"#$"%!%& .# a
'*
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%+) h(x) %4$ /5$%45! ),./'+",+0'0.6 7#$4$ ,. % .0&"5$"! 3'4 "#$ 4%",'+%5 30+/",'+ k(x) = x1 6
85$%45! "#$4$ ,. % 9%1 %" 0 2,"# '+$ .,)$ '3 "#$ 30+/",'+ 9',+9 ': "' −∞ %+) "#$ '"#$4 .,)$
9',+9 ': "' ∞6 ;' ," ,. /'44$/" "' .%! "#%" k(x) ,. ),./'+",+0'0. '($4 "#$ ,+"$4(%5 (−∞, ∞)
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.,+/$ 0 ,. +'" ,+ ,". )'<%,+6 7#,. ,. % <%""$4 '3 +'<$+/5%"04$ = <%>,+9 /5$%4 2#%" ,. /%55$)
2#%"6
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lim+ f (x) = f (a)6
@ 30+/",'+ "#%" ,. /'+",+0'0. %" $($4! 1',+" ,+ ,". )'<%,+ ,. /%55$) %
x→a
3&#%$#)&)1 ')#3%$&#6 @ 30+/",'+ ,. .%,) "' &$ 3&#%$#)&)1 &# /# $#%!-6/0 ,3 ," ,. &'"#
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/'+",+0'0. 34'< "#%" .,)$6C
x→a−
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7#$ )'<%,+ '3 "#,. 30+/",'+ ,. [0, ∞)6 * ;,+/$ ," ,. )$*+$) '+5! "' "#$ 4,9#" '3 0- ," )'$.+G"
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!"#$% &'(' 7#$ .H0%4$ 4''" 30+/",'+ f (x) =
√
x
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AFC I$<'(%&5$ ),./'+",+0,"!J 2#$+ "#$ 30+/",'+ #%. % #'5$ %" % 1',+" t- &0" %5.' #%. %
5,<," %" t- %. ,+ "#$ 1,$/$2,.$ 30+/",'+ 34'< "#$ .$/'+) $K%<15$
(
3x ,3 x 6= 2
g(x) =
8
,3 x = 2
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t
4,' -12 6(%(- '+2)
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h(x) =
(
1
−1
(3
(3
x≥0
x<0
:" ;,<,(-2 '()*+,-(,$(-./ 012, -12 3$,*-(+, =+2) -+
−∞
+, +,2 )('2 4,'
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k(x) =
!"#$% &'&'
∞
+, -12
1
x
>12 -1922 -.&2) +3 '()*+,-(,$(-(2)
?+6.,+%(46 3$,*-(+,) 4,' 27&+,2,-(46 3$,*-(+, 492 *+,-(,$+$) +,
(−∞, ∞)@
A4-(+,46
3$,*-(+,)8 6+= 3$,*-(+,)8 4,' -9(=+,+%2-9(* 3$,*-(+,) 492 *+,-(,$+$) +, -12(9 '+%4(,) B22&C
(,= (, %(,' -14- -12 '+%4(,) +3 94-(+,46 3$,*-(+,) 4,' -9(=+,+%2-9(* 3$,*-(+,) 492 (,-29C
9$&-2' D. E29-(*46 4).%&-+-2)"@
!"#$%&
!
' F$&&+)2 f (x) =
(
1
x
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3
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f (x)
'()*+,-(,$+$)@
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f (x)
() )1+0, (, H(=$92 !@:@
"
ID)29E2 -14- -1292 () 4 5$%&
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L46$2 >12+92%@ M2 0(66 $,'29)-4,' (- (,-$(-(E26. 1292 4,' &9+E2 (- (, K14&-29 N!@
)*+#,+- !.! /0(&+,-+1'2&+ 32$%+ &*+#,+-4 F$&&+)2 f
+,
[a, b]
4,' )$&&+)2
)$*1 -14-
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f (c) = w@
w
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f (a) ≤ w ≤ f (b)@
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c ∈ [a, b]
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f (x) =
(
sin x1
0
x 6= 0
x=0
<'$ %/6 */)"$D%? [a, b] +'/)%*/*/# J5 f (x) )%K"- '/ "D"$6 D%?." 7"),""/ f (a) %/3 f (b)
-'4",!"$" */ [a, b]1 L.) f *- /') +'/)*/.'.- %) J5 7"+%.-" limx→0 f (x) 3'"- /') "9*-)1
2- % +'$'??%$6 ," '7)%*/ )!" ('??',*/# $"-.?) ,!'-" &$''( *- "%-61 2 +**$ '( % (./+)*'/ f
*- % /.47"$ c -.+! )!%) f (c) = 01 M'')- %$" %?-' +%??"3 -%+*# '( )!" (./+)*'/1
.*+*//0+1 2343 56*/-0&*7# ()%*+%,8 ;.&&'-" f *- % (./+)*'/ )!%) *- +'/)*/.'.- '/ [a, b]
%/3 -.&&'-" f (a) %/3 f (b) !%D" '&&'-*)" -*#/-1 !"/ )!"$" "9*-)- % $'') '( f */ [a, b]1
2++'$3*/# )' )!" +'$'??%$65 *( f (a) *- /"#%)*D" %/3 f (b) *- &'-*)*D" %/3 ," +%/ 3$%, )!"
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!
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f (b). ?, #1& 89#&%7&:!+#& ;+*$& 01&(%&7> #1&%& !5 + 9$73&% c !9 [a, b] 5$"1 #1+# f (c) = 0.
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