limite de una funcion1
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limite y continuidad de una funcion - uapTRANSCRIPT
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Dr. Eberardo Osorio Rojas
MATEMATICA I
Módulo: 2 Unidad: I Semana:03
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LIMITE DE UNA FUNCION
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Lim f(x) = L
X a
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x→1- (x²-1)/(x-1) x→1+
Lim (x²-1)/(x-1) =
x →1
Lim (x-1)(x+1)/(x-1) =
x →1
Lim (x+1) = 1+1 = 2
x →1
Lim (x²-1)/(x-1) = 2
x →1
x→1- (x²-1)/(x-1)
0.9 1.9
0.99 1.99
0.999 1.999
1 0
x→1+ (x²-1)/(x-1)
1.1 2.1
1.01 2.01
1.001 2.001
2 2
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x→0- senx
x→0- senx
-0.1 -0.0998
-0.01 -0.00999
--0.001 - 0.00099
0
x→0+ senx
0.1 0.0998
0.01 0.0099
0.001 0.00099
0 0
Lim senx = 0
x →0
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Lim /x//x= -1
x→0-
x→0- IxI/x
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Y=x+1
Y = 2+1=3
Y= 4 +1 =5
Y= -x +2
Y= -(-1)+2
Y= 1+2=3
(0, 2)
(-1, 3)
Y=-x+2
f(1+) = lim (x+1) = 1+1 = 2
x →1+
f(1-) = lim (-x+2) = -1 +2 = 1
x →1-
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x→0- 1/x²
0 -1 1
Lim 1/x² =1/0 = ∞
x→0
x→0- 1/x²
-0.1 100
-0.01 10000
-0.001 1000000
x→0+ 1/x²
0.1 100
0.01 10000
0.001 1000000
0.0001 100000000
∞ ∞ Lim 1/x² =1/0 = ∞
x→0
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x→-∞ 1/x
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Lim f(x) =
x→∞
Lim (x+1)/(x+2) =
x→∞
Lim (1 +1/x) / (1+2/x)
x→∞
=(1+0)/(1+0)
= 1/1 =1
Lim f(x) = 1
x→∞
1/∞= 0
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Lim f(x) = Lim (x+1)(x-1) / (x-1) = Lim (x+1) = 1+1 = 2
x→1 x→1 x→1
Lim (x²-1)/(2x²-1) = Lim (1 – 1/x²) / (2 – 1/x² ) = (1 – 0)/ (2 – 0) = 1/2
x →∞ x →∞
Lim (x - √x²+1 )(x + √x²+1 ) / (x + √x²+1)= lim (x² - (x²+1)) / (x + √x²+1)
x →∞ x →∞
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(x²-1)/x² / (2x²-1)/x² =
(x²-1)/x² / (2x²-1)/x² =
(1 -1/x²) / (2 -1/x²) =
(1 – 0 ) / (2 - 0) =
1 / 2 =
(x - √x²+1 )( x + √x² +1) / ( x + √x² + 1 ) = x² - (√x²+1)² / ( x + √x² + 1 )
( x² - (x²+1)) / (x + √x² +1) = ( x² - x² -1 ) / (x + √x²+1 )= - 1/ (x + √x² +1)
(a - b)( a + b) = (a² - b²)
a² + ab – ba – b² = a² - b²
a=x b= √x²+1
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lim (x²-4)/(x-2)
x→2
2
4
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Lim x² = 4
x→2-
Lim x² = 4
x→2+
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0
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LIMITES INFINITOS
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Lim 1/ (1/x-1/x²) = 1/(0-0)=
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Ind
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Las indeterminaciones
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-1 1
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ASÍNTOTAS DE UNA FUNCIÓN
Las asíntotas son rectas a las cuales la función se va aproximando indefinidamente,
cuando por lo menos una de las variables (x o y) tienden al infinito. Una definición
más formal es:
DEFINICIÓN Si un punto (x,y) se desplaza continuamente por una función y=f(x) de
tal forma que, por lo menos, una de sus coordenadas tienda al infinito, mientras que
la distancia entre ese punto y una recta determinada tiende a cero, esta recta recibe
el nombre de asíntota de la función.
Las asíntotas se clasifican en:
Asíntotas verticales (paralelas al eje OY)
Si existe un número “a” tal, que :
La recta “x = a” es la asíntota vertical.
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Asíntotas horizontales (paralelas al eje OX)
Si existe el límite:
:
La recta “y = b” es la asíntota horizontal.
Ejemplo:
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Los valores de k hay que buscarlos entre los puntos que no pertenecen al
dominio de la función.
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(x²+2)/(x² - 2x)
(x²+2)/(x - 2) – x=
(x²+2)/(x - 2) – x(x-2)/(x-2)=
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2
-2
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GRACIAS