calculo- 27+56
TRANSCRIPT
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7/25/2019 calculo- 27+56
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ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
LIMITES
Calcular los siguientes lmites:
27. limx 0
1+x1x31+x
31x
a3b3=(ab )(a2+ab+b2 )
a=31+x ;b=31x
limx 0
1+x1x31+x
31x
=0
0
limx 0
1+x1x31+x
31x
= (1+x1x ) (1+x+1x )
[(31+x
31x )(
3
(1+x )2+
3
1 (x )2+
3
(1x )2)
3
(1+x)2+
3
1(x )2+
3
(1x )2 ] [(1+x+1x ) ]
1+x1+x
[ 1+x1+x3(1+x )2+ 31 (x )2+ 3(1x )2 ] [(1+x+1x ) ]lim
x0
limx0
2x
[ 2x3(1+x )2+ 31 (x )2+ 3(1x )2 ] [(1+x+1x ) ]limx0
1
[ 1+x+1x
3
(1+x )2+ 31 (x )
2+ 3(1x )2
]
=limx 0
1
lim
x 0
1+x+limx 0
1x
3 limx0
(1+x )2+ 3 lim
x0
1(x )2+ 3 lim
x0
(1x )2
=1
2
3
=3
2
28. limx 1
4x13x1
limx 1
4x13x1
=0
0
m.c .m (4,3 )=12
1
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ESCUELA POLITECNICA
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PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
y12=x {
4
x=y3
3
x=y4}
limx 1
4
x13
x1=lim
y 1
y31
y41
limy1
(y1 )(y2+y+1)(y1 ) (y+1 )(y2+1)
limy1
(y2+y+1)(y+1 )(y2+1)
= limy1 (y2
+y+1 )limy1
y+1limy1
y2+1
= 12+1+1
(1+1 )(1+1)=3
4
29. limx 1
5x13x1
m.c .m=(3,5 )=15
x=y15=
{
5x=y
3
3
x=y5
}limx 1
5x13x1
=limy1
y31
y51
limy1
y31
y51
= (y1 )(y2+y+1 )
(y1 )(y4+y3+y2+y+1)
limy1
(y2+y+1)
(y4
+y3
+y2
+y+1)
=limy1
(y2+y+1)
limy1 (y4
+y3
+y2
+y+1)
= 1
2+1+1
1
4
+13
+12
+1+1
=3
5
30. limx 1(
3x15x1 )
m.c .m=(3,5 )=15
x=y15={3
x=y5
5
x=y3}
2
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ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
limy1
y51
y31
=(y1 )(y4+y3+y2+y+1)
(y1 )(y2+y+1 )
limy1
(y4+y3+y2+y+1)(y2+y+1)
=limy1
(y4+y3+y2+y+1)
limy1
(y2+y+1) =
14+13+12+1+1
12+1+1
=5
3
31. limx 0
31+x
31x
1+x1x
a3b3=( ab )(a2+ab+b2 )a=31+x ;b=
31x
limx 0
31+x
31x
1+x1x=
( 31+x31x )(3
(1+x )2+
3
1(x )2+
3
(1x )2) (1+x+1x)
(1+x1x ) (1+x+1x )(3
(1+x )2+
3
1(x )2+
3
(1x )2)
limx0
(1+x1+x )(1+x+1x )
(1+x1+x )( 3(1+x )2+ 31 (x )2+ 3(1x )2 )
limx 0
1+x1+x1+x1+x
1+x+1x
3
(1+x)2+
3
1(x )2+
3
(1x )2=
x
x1+x+1x
3
(1+x)2+
3
1(x )2+
3
(1x )2
limx 0
11+x+1x
3
(1+x )2+
3
1(x )2+
3
(1x )2=limx 0 1
limx 0
1+ limx 0
x+limx 0
1limx 0
x
3 limx 0
(1+x)2+ 3 limx 0
1(x )2+ 3 limx 0
(1x )2= 1
+1
1+1+1=2
3
32. limx a
2x2x2+2a2
xa
limx a
(2x2x2+2a2 ) . (2x+2x2+2a2 )(x+a)(xa ). (2x+2x2+2a2 )(x+a)
= limx a
( 4x22x22a2 )(x+a )
(xa ) (2x+2x2+2a2 )
3
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ESCUELA POLITECNICA
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Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
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04
limx a
(2x22a2 )(x+a)
(xa ) (2x+2x2+2a2)=
limx a
2 . limx a
(x+a ) . limx a
(xa ). limx a
(x+a)
limx a
(xa ) . limx a
(2x+2x2+2a2 )
2 ((a)+a )((a)+a)(2(a)+2(a)2+2a2)
=4 a(2a)2a+2a
=2a
33. limx a
xa
2x2x2
+2a2
limx a
(xa ) . (2x+2x2+2a2 )(x+a )(2x2x2+2a2 ) . (2x+2x2+2a2 )(x+a)
= limx a
(xa ) (2x+2x2+2a2 )( 4x22x22a2 )(x+a )
limx a
(xa ) (2x+2x2+2a2)( 2x22a2 )(x+a)
=limxa
(xa ) . limx a
(2x+2x2+2a2 )
limxa
2 . limxa
(x+a ) . limxa
(xa ). limx a
(x+a)
limx a
(2(a)+2(a)2+2a2 )2 ((a)+a )((a)+a)=
2a+2a4a (2a)=
1
2a
34. limx
1x+3x
lim
x
1x+3
x=lim
x1x
3
x
+3
x3
x
= limx
(
3
1
x
1
)+1
1= (01 )+1=0
limx
(3 1x1)+1=[3 limx 1x limx 1]+ limx
4
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ESCUELA POLITECNICA
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PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
35.x
2+2+2x
limx
)
x2+2+2x
()=x2
x2+2
x2+
2x
x2=1+
2
x +2
limx
limx
1+ 2
x +2= lim
x1+( lim
x2 limx
1
x )+ limx 2=1+0+2=3
36. limx 0
10
x+11x
limx 0
10x+11
x =
limx 0
10x+11
l x0
x =
10
limx0
(x+1)limx0
1
limx0
x =
10(0+1)10
=1011
0 =
110 =
0
0
limx 0
10
x+11x
=limx 0
10
x+11x
10x+1+1
10
x+1+1=lim
x 0
(10x+1 )2
1
x (10x+1+1 ) (x+1 )=y
limx0
5
(y )11
x (10y+1)
5
(y )4+
5
(y )3+
5
(y )2+
5
(y )1+1
5
(y )4+ 5(y )
3+ 5(y )2+ 5(y )
1+1=
[(5(y )15(y )4 )+(5(y )15(y )3 )+(5(y )15(y )2)+(5(y )15(y )1 )+(5(y )1 ) ][5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1 ]x ( 10y+1)[5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
=
limx0
limx 0
y+( 5(y )4 )+(5
(y )3 )+( 5(y )2)+(
5
(y )1 )5(y )4
5
(y )3
5
(y )2
5
(y )11
x ( 10y+1 )[5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1 ] =
5
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ESCUELA POLITECNICA
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PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
limx 0
y1
x (10y+1 )[5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]y= (x+1 )
limx 0
x+11
x (10x+1+1 )[ 5(x+1 )4+ 5(x+1 )3+ 5(x+1 )2+ 5(x+1 )1+1 ]=
x
x (10x+1+1 )[5(x+1 )4+ 5(x+1 )3+ 5(x+1)2+ 5(x+1 )1+1 ]=
limx 0
limx 0
1
(10x+1+1)[5(x+1 )4+ 5(x+1 )3+ 5(x+1 )2+ 5(x+1 )1+1 ]=
limx0
1
limx0
(10x+1+1)[5(x+1 )4+ 5(x+1 )3+ 5(x+1 )2+ 5(x+1 )1+1 ]=
limx 0 1
limx 0
(10x+1+1 )[ limx 05
(x+1)4+
5
(x+1 )3+
5
(x+1 )2+
5
(x+1)1+1]
=
limx 0
1
(limx0
10x+1+1)[ 5 limx0 (x+1 )
4+
5
lim x 0 (x+1 )3+
5
lim x 0 (x+1 )2+
5
lim x 0 (x+1 )1+limx 0
1]=
1
( 100+1+1 )[
5
(0+1 )4+5
(0+1 )3+5
(0+1 )2+5
(0+1 )1+1
]
=
1
( 101+1)[5(1 )4+ 5(1 )3+ 5(1 )2+ 5(1 )1+1 ]=
1
(1+1 )[1+1+1+1+1 ]=
1
(2 )[5 ]= 1
10
37. limx a
2x2x2+2a2
xa
6
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ESCUELA POLITECNICA
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PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
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04
limx a
(2x2x2+2a2 ) . (2x+2x
2+2a2 )(x+a)(xa ). (2x+2x2+2a2 )(x+a)
= limx a
( 4x22x22a2 )(x+a )(xa ) (2x+2x2+2a2 )
limx a
( 2x22a2 )(x+a)
(xa ) (2x+2x2+2a2)=
limxa
2 . limxa
(x+a ) . limxa
(xa ). limx a
(x+a)
limxa
(xa ) . limx a
(2x+2x2+2a2 )
limx a
2 ((a)+a )((a)+a)(2(a)+2(a)2+2a2)
=4 a(2a)2a+2a
=2a
38. limx 1
x7+5x6+4x3
x7+2x3
x
limx1
( 4)+ limx1
(5x3)+ limx1
(4)
limx1
(x4 )+ limx1
(2 )
x4
+5x3
+4x
4+2=
x3 (x4+5x3+4 )x
3 (x4+2) = lim
x1
limx1
= (1)+5 (1)3+4
(1)4+2=
4
2=2
39. limx 1
x
4
x3
+x2
3x+2x
3x2x+1
x4x3+x23x+2
x3x2x+1
= limx1
x+ 2x
24x+2
x3x2x+1
= limx 1
x+ 2 (x1 )2
(x1 )2 (x+1 )=
limx1
7
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Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
x+limx1
2
l x1
(x+1 )=1+
2
(1+1 )=1+
2
2=1+1=2
x+ 2
(x+1 )= lim
x1
limx 1
40. limx 1
1 3x
1 5x
m.c .m (3,5)=15
x=z15{3
x=z5
5
x=z3}
1z5
1z3= lim
x1
z51
z31
=(z1 )(z4+z3+z2+z+1)
(z1 )(z+1)limx 1
limx 1
(z4+z3+z2+z+1)
(z+1) =
limx 1
(z4+z3+z2+z+1)
limx1
(z+1) =
14+13+12+1+1
1+1 =
5
2
41. limx 1[x
51
x41 ]
(x1 )(x4+x3+x2+x+1)
(x1 )(x3+x2+x+1)=
limx 1
(x4+x3+x2+x+1)
limx1
(x3+x2+x+1)
limx1
8
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ESCUELA POLITECNICA
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Realizad !":
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Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
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04
xxx
limx 1
(2)+ limx 1
(x)+limx 1
1
x
limx 1
(x3)+ limx 1
( 2)+limx 1
(x)+limx 1
(1)
(4 )+lim
x 1
(3)+
lim
x 1
= (1)4+(1)3+(1)2+(1)+1
(1)3+(1)2+(1)+1=
5
4
42. limx 1[ 31x3 + 1x1 ]
limx 1[ 31x3 11x ] = limx 1[31(1+x+x
2)
1x3 ]=limx 1[31xx2
1x3 ] lim
x 1[(x
2
+x2)1x3 ]
=limx 1[
(x+2 )(x1)(1x )(1+x+x2) ]= lim
x 1[ (x+
2
)(1
x)(1x )(1+x+x2) ] lim
x 1
x+2
1+x+x2=
limx1
(x+2 )
limx 1
(1+x+x2 )=
(1 )+(2 )1+(1 )+(1 )
=3
3=1
43. limx 0
21+x+x22xx2
limx 0
21+x+x2(2+x )x
2 =
limx0 (21+x+x
2(2+x )) (21+x+x2+(2+x ) )
x2 (21+x+x2+(2+x ) )
21+x+x2
limx 0
4+4x+4x244xx2
x2 (21+x+x2+ (2+x ) )
=lim
x 0
3x2
x2 (21+x+x2+ (2+x ) )
=limx 0
(3)
limx 0
(+ limx 0
(2+x ))
3
(2
1+(0)+(0)2)+ (2+(0))
= 3
4
9
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ESCUELA POLITECNICA
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PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
44. limx 0
5
2x2+10x+17x
2+10x+1x
limx 0
52x2+10x+17x2+10x+1x
=lim
x 0
52x2+10x+1 7x2+10x+1
limx 0
x=
5
limx0
2x2+10x+17lim
x0
x2+10x+1
limx0
x =
5
2 (0 )2+10 (0 )+1
7
(0 )2+10 (0 )+1
0
50+0+1
70+0+1
0 =
51
71
0 =
110 =
0
0
limx 0
52x2+10x+17x2+10x+1x
= limx0
52x2+10x+11( 7x2+10x+11)x
52x2+10x+11x
7x2+10x+11
x x
2+10x+1=w;2x2+10x+1=y
limx 0
5y1x
7w1x
limx0
[ 5y1x
5
(y )4+
5
(y )3+
5
(y )2+
5
(y )1+1
5
(y )4+
5
(y )3+
5
(y )2+
5
(y )1+1 ][
7w1x
7
(w )6+
7
(w )5+
7
(w )4+
7
(w )3+
7
(w )2+
7
(w )1+1
7
(w )6+
7
(w )5+
7
(w )4+
7
(w )3+
7
(w )2+
7
(w )1+1 ]
limx0
limx 0
[(5(y )15(y )4 )+(5(y )15(y )3)+(5(y )15(y )2)+(5(y )15(y )1 )+(5(y )1 )][5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]x [ 5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
[(7(w )17(w )6 )+(7(w )17(w )5 )+(7(w )17(w )4 )+(7(w )17(w )3)+(7(w )17(w )2)+(7(w )17(w )1 )+(7(w )1 )] [7(x [ 7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
10
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ESCUELA POLITECNICA
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Ing. Ezequiel Guamn
Realizad !":
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Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
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04
limx 0
y+ 5(y )4+ 5(y )
3+ 5(y )2+ 5(y )
15(y )4 5(y )
35(y )2 5(y )
11
x [5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1 ]
w+7
(w )6+
7
(w )5+
7
(w )4+
7
(w )3+
7
(w )2+
7
(w )1
7
(w )6
7
(w )5
7
(w )4
7
(w )3
7
(w )2
7
(w )11
x [ 7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
limx 0
y1
x
[
5
(y )4+5
(y )3+5
(y )2+5
(y )1+1
]
w1
x
[
7
(w )6+7
(w )5+7
(w )4+7
(w )3+7
(w )2+7
(w )1+1
]x2+10x+1=w ;2x2+10x+1=y
limx 0
2x2+10x+11
x [ 5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
x2+10x+11
x [ 7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
limx 0
2x2+10x
x [ 5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
x2+10x
x [ 7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
limx 0
x (2x+10)
x [ 5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
x (x+10)
x [ 7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
limx 0
(2x+10)
[5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
(x+10)
[7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
2x2+10x+1=y ; x2+10x+1=w
limx 0
2x2+10x+1=2 (0 )2+10 (0 )+1=1 limx 0
y=1 ;
limx 0
x2+10x+1=(0 )2+10 (0 )+1=1 lim
x0
w=1
limx 0
(2x+10)
[5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
(x+10)
[7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
11
-
7/25/2019 calculo- 27+56
12/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
limx0
(2x+10)
limx0
[5(y )4+ 5(y )3+ 5(y )2+ 5(y )1+1]
limx0
(x+10)
limx0
[ 7(w )6+ 7(w )5+ 7(w )4+ 7(w )3+ 7(w )2+ 7(w )1+1]
2(0)+10
[ 5(1 )4+ 5(1 )3+ 5(1 )2+ 5(1 )1+1]
0+10
[7(1 )6+ 7(1)5+ 7(1 )4+ 7(1 )3+ 7(1 )2+ 7(1 )1+1]
0+10
[1+1+1+1+1 ]
10
[1+1+1+1+1+1+1 ]=
10
5
10
7
2
1
10
7
=1410
7
=4
7
45. limx 0
m1+ax
n1+bx
x
m1+ax
n1+bx
x =
limx 0
m1+ax
n1+bx
limx 0
x =
m
limx 0
(1+a x) nlimx 0
(1+bx)
limx 0
x
limx 0
m(1+a(0))n(1+b (0))
0 =
m(1+0) n(1+0)0
=m1
n1
0 =
110 =
0
0
limx 0
m1+ax
n1+bx
x =lim
x0
(1+ax )1
m
x
(1+bx )1
n
x
limx 0[(1+ax )
1
m
x (1+ax )
m1m +(1+ax)
m2m + (1+ax )
1
m+1
(1+ax )m1
m + (1+ax )m2
m +(1+ax)1
m+1 ][(1+b x )
1
n
x (1+bx )
n1n +(1+bx )
n2n +(1+bx )
1
n
(1+bx )n1
n + (1+bx )n2
n + (1+bx )1
n+1
limx 0 [
ax
x ( (1+ax )m1m +(1+ax)
m2m + (1+ax )
1
m+1)][ bx
x ( (1+bx )n1n +(1+bx )
n2n + (1+bx )
1
n+1)]=
limx 0 [
a
((1+ax )m1
m +(1+ax )m2
m + (1+ax )1
m+1)][ b
((1+bx )n1
n + (1+bx )n2
n + (1+bx )1
n+1)]=
12
-
7/25/2019 calculo- 27+56
13/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
limx0
a
limx0
( (1+ax )m1m +(1+ax )
m2m + (1+ax )
1
m+1)
limx0
b
limx0
( (1+bx )n1n + (1+bx )
n2n + (1+bx )
1
n+1)=
a
((1+(0)a )m1
m +(1+a (0))m2
m + (1+a(0))1
m+1)
b
((1+b(0))n1
n + (1+b(0))n2
n + (1+b (0))1
n+1)=
a
((1+(0)a )m1
m +(1+a (0))
m2
m + (1+a(0))
1
m+1)
b
((1+b(0))n1
n + (1+b(0))
n2
n + (1+b (0))
1
n+1)
=
a
( (1+0 )m1
m + (1+0)m2
m + (1+0 )1
m+1)
b
( (1+0 )n1
n +(1+0 )n2
n + (1+0 )1
n+1)=
a
( (1 )m1m +(1 )
m2m + (1)
1
m+1)
b
( (1 )n1n + (1 )
n2n + (1 )
1
n+1)=
a
( (1 )m1m + (1 )
m2m + (1 )
1
m+1)
b
((1)n1n + (1 )
n2n +(1 )
1
n
a
(1+1+1+1 )
b
(1+1+1+1 )=
a
m
b
n=
anbmnm
46. limx4
35+x15x
limx4
35+x15x
=limx 4
35+x
limx 4
15x=
limx4
3limx4
5+x
limx4
1limx 4
5x=
35+4154
=3911
=3311
=0
0
13
-
7/25/2019 calculo- 27+56
14/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
(4x )(1+5x )(4+x )(3+5+x )
=
limx 4
35+x
15x=
limx 4
35+x15x
3+5+x
3+5+x=lim
x 4
9(5+x)
(15x ) (3+5+x )=lim
x 4
95x(15x) (3+5+x )
=limx 4
(1
( 4x )(1+5x )( 4x )(3+5+x )
=limx4
(1+5x )(3+5+x )
=limx 4
1+5x
limx 4
35+x
limx 4
=
limx4
1+limx4
5x
limx4
3limx 4
5+x=
1+5435+4
= 1+139
= 1+133
= 2
6=1
3
47. limx 3
x22x+6x2+2x6x24x+3
limx 3
x22x+6x2+2x6x
2
4x+3
limx3
x22x+6lim
x3
x2+2x6
limx3
x24x+3
=3
22 (3 )+632+2 (3 )6
324 (3 )+3
=99
1212=
0
0
limx 3
x22x+6x2+2x6x24x+3
limx3
x22x+6x2+2x6(x3 ) (x1 )
x22x+6+x2+2x6
x22x+6+x2+2x6 =
limx 3
(x22x+6)(x2+2x6 )
(x3 ) (x1 ) (x22x+6+x2+2x6 )=lim
x3
x22x+6x22x+6
(x3 ) (x1 )(x22x+6+x2+2x6 )=
limx 3
4x+12
(x3 ) (x1 ) (x22x+6+x2+2x6 )=lim
x3
4 (x3 )
(x3 ) (x1 )(x22x+6+x2+2x6 )=
limx 3
4
(x1 )(x22x+6+x2+2x6)= 4
(31 )(322 (3)+6+32+2 (3 )6)=
14
-
7/25/2019 calculo- 27+56
15/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
4
(2 )(9+9)= 4
(2 ) (3+3 )= 4
(2 ) (6 )=4
12=1
3
48. limn
n (a1 ) ; a>0
n (a1 )=limn
n (a1 )1
(a+1)
(a+1 ) = lim
n
n (a1 )
(a+1 )=
limn
n (a1 )n
(a+1)n
=limn
(a1 )
( an2+ 1n )=
limn
(a1 )
limn( an2+ 1n )
=limn
(a1 )
(limn
a
n2 +
limn
1
n)=
limn
(a1)(0+0 )
= (a1 )
0 =
49. limx
(x+1 )2 (37x )2
(2x1 )4
limx
(x+1 )2 (37x )2
(2x1)4 = lim
x
(x+1 )2 (37x )2
x4
(2x1 )4
x4
= limx
(x+1 )2
x2(37x )2
x2
( 2x1x )4 =
limx
(x+1x )
2
(37xx )
2
( 2xx 1x )4 = lim
x
(x
x+ 1x )
2
(3
x7x
x)2
(21x )4
= limx
(1+1
x )2
(3
x7)
2
(21x )4 = lim
x
(1+0 )2 (07 )2
(20 )4 =
limx
(1 )2 (7 )2
(2 )4 =lim
x
(1 ) (49 )16
=49
16
50. limx
x2+8x+3x2+4x+3
Si t=-x
t
15
-
7/25/2019 calculo- 27+56
16/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
(t28 t+3t24 t+3)=
limt
(t28 t+3t
24 t+3) (t28 t+3+t
24 t+3)
(t28 t+3+t24 t+3 )limt
limt
t28 t+3t2+4 t3
t28 t+3+t24 t+3=
limt
4 t
t28 t+3+t24 t+3=
limt
4
18t+ 3t2+14t+3t2=2
51. limx
(1+x11+7x13 )3
(1+x4 )10
(1+x11+7x13)3
x40
(1+x4 )10
x40
=limx
(1+x11+7x13)3
x39
1
x
(1+x4
x4
)10
=
(1+x11+7x13)3
(1+x4 )10 = lim
x
limx
( 1x13+x
11
x13+7x
13
x13 )
3
1
x
(1
x4 +1)10
=limx
( 1x13+1
x2+7)
3
1
x
(1
x4 +1)10
=
(1+x11+7x13
x13 )
3
1
x
( 1x4 +x
4
x4 )
10 = lim
x
limx
16
-
7/25/2019 calculo- 27+56
17/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
(7 )30
(1)10 =
0
1=0
( 0+0+7 )30
(0+1 )10 = lim
x
limx
52. limx
3
1+
4
x
4
1+
3
x
1515xlimx
31+ 4x41+ 3x1515x
=
3 limx 1+ 4x 4 limx 1+ 3xlimx
15 limx 15x=
31+0
41+0
1510
=31
41
151
=
1111
=0
0
limx
3
1+ 4
x4
1+ 3
x
1515x=lim
x
(3
1+ 4
x1)
(4
1+ 3
x1)
1515x=
limx
[
31+ 4x11
3(1+ 4x )
2
+3(1+ 4x )
1
+1
3
(1+
4
x
)
2
+3
(1+
4
x
)
1
+1
]
[
41+ 3x11
4(1+ 3x )
3
+4(1+ 3x )
2
+4(1+ 3x )
1
+1
4
(1+
3
x
)
3
+4
(1+
3
x
)
2
+4
(1+
3
x
)
1
+1
]1515x1
5(15x )
4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+1
5(15x )4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+1
=
17
-
7/25/2019 calculo- 27+56
18/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
( 31+ 4x3(1+ 4x)2
)+( 31+ 4x3(1+ 4x )1
)+31+ 4x 3(1+ 4x )
2
3(1+ 4x )
1
1
3(1+ 4x )2
+3(1+ 4x )
1
+1
[( 41+ 3x4(1+ 3x )
3)+( 41+ 3x4(1+ 3x )2)+( 41+ 3x4(1+ 3x )
1)+ 41+ 3x4(1+ 3x )3
4(1+ 3x )
2
4(1+ 3x )
1
1
4
(1+ 3x )
3
+ 4
(1+ 3x )
2
+ 4
(1+ 3x )
1
+1
]lim
x
limx [
1+4
x+
3(1+ 4x)2
+ 31+ 4x3(1+ 4x )2
3(1+ 4x )
1
1
3
(1+ 4
x )2
+ 3
(1+ 4
x )1
+1
]
[1+
3
x+
4(1+ 3x )3
+4(1+ 3x )
2
+ 41+ 3x4(1+ 3x )3
4(
4
(1+ 3
x )3
+ 4
(1+ 3
x )2
+ 4
(1+ 3
x )1
+
5(15x )4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+11+5
x
5(15x )4
5(15x )
3
5(15x )
2
5(
5(15x )4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+1
limx [
1+4
x1
3
(1+4
x )2
+3
(1+4
x )1
+1
]
[ 1+
3
x1
4
(1+3
x )3
+4
(1+3
x )2
+4
(1+3
x )1
+1
]+11+ 5x5(15x )
4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+1
=
18
-
7/25/2019 calculo- 27+56
19/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
limx
[ 4
x
3(1+ 4x )2
+3(1+ 4x )
1
+1 ][ 3
x
4(1+ 3x )3
+4(1+ 3x )
2
+4(1+ 3x )
1
+1 ]5
x
5(15x )4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+1
= limx
1
x[
4
3(1+ 4x )2
+3(1+ 4x )
1
+1 ][ 4(11
x[ 5(15x )4+ 5(15x )3+
limx
[ 4
3(1+ 4x )2
+3(1+ 4x )
1
+1 ][ 3
4(1+ 3x )3
+4(1+ 3x )
2
+4(1+ 3x )
1
+1 ]5
5(15x )4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+1
=
limx [
4
3(1+ 4x )2
+3(1+ 4x )
1
+1 ] limx [ 4(lim
x
5
5(15x )4
+5(15x )
3
+5(1
[ lim
x
4
limx
3(1+ 4x )2
+3(1+ 4x )
1
+1 ]
[ lim
x
3
limx
4(1+ 3x )3
+4(1+ 3x )
2
+4(1+ 3x )
1
+1 ]limx
5
limx
5(15x )4
+5(15x )
3
+5(15x )
2
+5(15x )
1
+1
= [ lim
x
4
3 limx (1+ 4x )2
+3 limx (1+ 4x )
1
+ limx
5 limx (15x )4
+5 limx (
[ 4
3
(1+0 )2+ 3(1+0 )
1+1
]
[ 3
4
(1+0 )3+ 4(1+0 )
2+ 4(1+0 )1+1
]55
(10 )4
+5
(10 )3
+5
(10 )2
+5
(10 )1
+1
= [ 4
3
(1 )2+ 3(1 )
1+1
]
[ 3
4
(1 )3+ 4(1 )
2+ 4(1 )1+1
]55
(1 )4
+5
(1 )3
+5
(1 )2
+5
(1 )1
+1
=
[ 41+1+1 ][ 31+1+1+1 ]5
1+1+1+1+1
=[43 ][34 ]
5
5
=[16912 ]
1 =
[ 712 ]1 =
7
12
19
-
7/25/2019 calculo- 27+56
20/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
53. l x
x
x+x+x
limx
x
x+x+x= lim
x
x
x
x+x+xx
=limx
1
xx + x+xx=
limx
1
1+1
x+ x
x2
=lim
x
1
1+1
x+
1
x3
=
limx
1
limx 1+ limx 1x + limx 1x3 =
1
1+0+0=1
54.
x25x+6(x)l x
x25x+6
(x)=limx
(x25x+6x)(x25x+6+x)
(x25x+6+x )limx
limx+
(x25x+6x2)
(x25x+6+x)=
limx+
(65x)
(x25x+6+x)=
limx+
65xx2
x25x+6+xx2
=
limx+
6x2
5xx2
x25x+6+xx2
=
Cuando x +|x2|=x
=
limx+
6
x5
x
2
x25x
x2+ 6
x2 +xx
=limx+
6
limx+
1
x lim
x+5
limx+
1 limx+
5limx+ 1
x + lim
x+6
limx+ 1
x2 + lim
x+1
=
20
-
7/25/2019 calculo- 27+56
21/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
=6(0)5
15 (0 )+6(0)+(1)=
5
2
Cuando x |x2|=x
=
limx
6x +5
x2
x2
5x
x2+
6
x2
x
x
=lim
x 6
limx
1
x + lim
x 5
limx 1 limx 5lim
x 1
x + lim
x6
limx
1
x
2 lim
x1
=
=6 (0 )+5
15 (0 )+6(0)(1)=
5
0=
55.
x2+1x (x )lim
x
x2
+1
x (x )=lim
x
x (x2+1x )(x2+1+x)
(x2+1+x)lim
x
limx+
x (x2+1x2)
(x2+1+x )=
limx+
x
(x2+1+x )=
limx+
x
x2
x2+1x2
+ x
x2=
Cuando x +|x2|=x
=
limx+
1
1+ 1x2 +1=
limx+
1
limx+ 1+lim
x+1
x2 + lim
x+1
=
1
1+(0 )+(1)=
1
2
Cuando x |x2|=x
21
-
7/25/2019 calculo- 27+56
22/23
ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
=
limx
1
1+ 1x21=
limx
1
limx 1+lim
x1
x2 lim
x1
=
1
1+(0 )(1)=1
0=
56.
x+31x2()
l x
x+31x2
1x2
x+ 3
(x2x31x2+
3
(1x2 )2
)
()=limx
limx
limx +
x2+1x2
x2x
31x2+3
(1x2 )2
=
limx+
1
x2x
31x2+3
(1x2 )2=
limx+
13x6
x2
3x6+x
31x23x6
+
3
(1x2 )2
3x6
=
Cuando x +|3x6|=x2
=
limx+
1
x2
x2
x2+1
x limx+
31x2+ 3(1x2)2
x6
=
limx+
1
x2
limx+
1+lim
x+1
x
limx+
3
1x2+ limx+
3(1x2 )2
x6
=
0
1+0+0=0
Cuando x |3x6|=x2
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7/25/2019 calculo- 27+56
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ESCUELA POLITECNICA
NACIONAL
PARALELO: GR3
CALCULO EN UNA VARIABLE
Ing. Ezequiel Guamn
Realizad !":
#uan Ca""e"a
Ca"l$ R%&a
#aime Va"ga$
'e%&a: ()*)+Se!,+()
N. de !gina$: *-
T"aa/ N.
04
=
limx
1
x2
x2
x2
1
x limx
31x2+ 3(1x2 )2
x6
=
limx
1
x2
limx
1+lim
x1
x lim
x
31x2+ limx
3(1x2 )2
x6
=
0
1+0+0=0
23