Cálculo1: 6 º Lista de Exercícios
Autor: Resuminho | Publicado: 2012-04-07T13:45:00-07:00


1)Calcule os seguintes limites:a)lim x =&nbsp; x®2b)lim x³ =&nbsp; x®2c)lim 2x =&nbsp; x®1d)lim x6&nbsp;=&nbsp; x®1e)lim (x²-x) =&nbsp; x®3f)lim (x+1)/x² =&nbsp; x®3g)lim (3x+x²) =&nbsp; x®2h)lim (x²+2x)/(3x-1) =&nbsp; x®1i)lim (4x²-2x+1) =&nbsp; x®3j)lim (2x²+3x²-x+3) =&nbsp; x®1l)lim (3x²-2x-1) =&nbsp; x®1m)lim (x+1)/x² =&nbsp; x®2n)lim (4x³+2x²+x+2) =&nbsp; x®0o)lim (x4-x³+x²+x+1) =&nbsp; x®1p)lim (-2x²-x+2)&nbsp;=&nbsp; x®-1q)lim (x³-x²-3 )10&nbsp;&nbsp;=&nbsp; x®2r)lim (x+2)5&nbsp;&nbsp;=&nbsp; x®1s)lim (x³+x²+x+1)/(x²-1) =&nbsp; x®3t)lim (x²+x)/(x³+x²+x+1) =&nbsp; x®0u)lim 3/(2x+1) =&nbsp; x®1v)lim (3x²+x+1)/2 =&nbsp; x®-1x)lim (x³+2x²+3x+2)/(2x³+x²+2x+4) =&nbsp; x®0
2)Calcule os seguintes limites indeterminados:a)lim (x²-9)/(x-3) =&nbsp; x®3b)lim (49-x²)/(7+x) =&nbsp; x®-7c)lim (5-x)/(25+x²) =&nbsp; x®5d)lim (x²+x)/(x²-3x) =&nbsp; x®0e)lim x³/(2x²-x) =&nbsp; x®0f)lim (49+14x+x²)/(7+x) =&nbsp; x®-7g)lim (x²-6x+9)/(x-3) =&nbsp; x®3h)lim (x²-4x+3)/(x-1) =&nbsp; x®1i)lim (x²-7x+12)/(x-4) =&nbsp; x®4j)lim (x-1)/(x²-3x+2) =&nbsp; x®1l)lim (x²-2x+1)/(x-1) =&nbsp; x®1m)lim (x-2)/(x²-4) =&nbsp; x®2
3) Calcule os seguintes limites indeterminados:a)lim -x =&nbsp; x®-4b)lim x³=&nbsp; x®-4c)lim -x4=&nbsp; x®-3d)lim 3x8=&nbsp; x®-1e)lim (-x²-3x) =&nbsp; x®5f)lim (3x+2)/x²=&nbsp; x®-2g)lim (8x+x²)=&nbsp; x®-1h)lim (5x-x²)/(x-3)=&nbsp; x®2i)lim (x²-2x-15)=&nbsp; x®-3j)lim (2x³+3x²-7x+3)=&nbsp; x®-1l)lim (4x³-x²-4x²+1)=&nbsp; x®1m)lim (x³-2x²-3x)/(x²-8)=&nbsp; x®3n)lim (4x³+2x²+x+2)=&nbsp; x®-1o)lim (x4-x³+x²+x+1)=&nbsp; x®0p)lim (-2x²-x+2)=&nbsp; x®3q)lim (x³-x²-3)7=&nbsp; x®-1r)lim (x²+1)4=&nbsp; x®2s)lim (7x8-x7)/(-2x³+5x²-x+6)=&nbsp; x®0
3) Calcule os seguintes limites indeterminados:a)lim (x²-16)/(x+4)=&nbsp; x®-4b)lim (64-x²)/(8-x)=&nbsp; x®8c)lim (9-x)/(81-x²)=&nbsp; x®9d)lim (3x²+x)/(x²-7x)=&nbsp; x®0e)lim x5/(5x²-3x)=&nbsp; x®0f)lim (16+8x+x²)/(4+x)=&nbsp; x®-4g)lim (x²-20x+100)/(x-10)=&nbsp; x®10h)lim (x²-9x+20)/(x-5)=&nbsp; x®5i)lim (x²+13x+42)/(x+6)=&nbsp; x®-6j)lim (x-4)/(x²-7x+12)=&nbsp; x®4k)lim (x²-8x+7)/(x-1)=&nbsp; x®1l)lim (x²-8x+7)/(x-1)=&nbsp; x®1m)lim (x-11)/(x²-121)=&nbsp; x®11
3) Calcule os limites abaixo:a)lim (x²+2x-1)=&nbsp; x®-2b)lim (y³-2y²+3y-4)=&nbsp; y®-1c)lim (t²-5)/(2t³+6)=&nbsp; t®2d)lim (2x+1)/(x²-3x+4)=&nbsp; x®-1e)lim (y³+8)/(y+2)=&nbsp; y®-2f)lim (s²-1)/(s-1)=&nbsp; s®1g)lim (x²+5x+6)/(x²-x-12)=&nbsp; x®-3h)lim Raíz quadrada de((8r+1)/(r+3))=&nbsp; h®1i)lim Raíz quadrada de((x²+3x+4)/(x³+1))=&nbsp; h®2j)lim&nbsp;Raíz quadrada de(((y²-9)/(2y²+7y+3))=&nbsp; y®-3k)lim&nbsp;Raíz quadrada de(((8t³-27)/(4t²-9))=&nbsp; t®3/2l)lim (raiz quadrada de(x+2)-raiz quadrada de 2)/x=&nbsp; x®0m)lim (2-raiz quadrada de (4-t))/t=&nbsp; t®0n)lim (raiz cúbica de (h+1)-1)/h=&nbsp; h®0o)lim (x³-x²-x+10)/(x²+3x+2)=&nbsp; x®-2p)lim (2x³-5x²-2x-3)/(4x³-13x²+4x-3)=&nbsp; y®3-------------------------------------------------------------------------------------Respostas
Cálculo1: 6 º Lista de Exercícios
1)Calcule os seguintes limites:a)lim x = 2&nbsp; x®2
b)lim x³ = 2³ =&nbsp;8&nbsp; x®2
c)lim 2x = 2(1) =&nbsp;2&nbsp; x®1
d)lim x6&nbsp;= 16&nbsp;= 1&nbsp; x®1
e)lim (x²-x) = 3²-3 = 9-3 = 6&nbsp; x®3
f)lim (x+1)/x² = (3+1)/3² = 4/9&nbsp; x®3
g)lim (3x+x²) = 3(2)+2² = 6+4 = 10&nbsp; x®2
h)lim (x²+2x)/(3x-1) = (1²+2(1))/(3(1)-1) = (1+2)/(3-1) = 3/2&nbsp; x®1
i)lim (4x²-2x+1) = 4(3²)-2(3)+1 = 4(9)-6+1 = 36-5 = 31&nbsp; x®3
j)lim (2x²+3x²-x+3) = 2(1²)+3(1²)-1+3 = 2+3+2 = 7&nbsp; x®1
l)lim (3x²-2x-1) = 3(1²)-2(1)-1 = 3-2-1 = 0&nbsp; x®1
m)lim (x+1)/x² = (2+1)/2² = 3/4&nbsp; x®2
n)lim (4x³+2x²+x+2) = 4(0)³+2(0)²+0+2 = 0+0+0+2 = 2&nbsp; x®0
o)lim (x4-x³+x²+x+1)&nbsp;=&nbsp;(14-1³+1²+1+1) = 1-1+1+1+1 = 3&nbsp; x®1
p)lim (-2x²-x+2)&nbsp;= -2(-1)²-(-1)+2 = -2(1)+1+2 = -2+3 = 1&nbsp; x®-1
q)lim (x³-x²-3 )10&nbsp;&nbsp;= (2³-2²-3)10&nbsp;=&nbsp;&nbsp;(8-4-3)10&nbsp; =&nbsp;&nbsp;(1)10&nbsp;= &nbsp;1&nbsp; x®2
r)lim (x+2)5&nbsp;&nbsp;= &nbsp;(1+2)5&nbsp;=&nbsp;(3)5&nbsp;&nbsp;= 81*3 = 243&nbsp;&nbsp; x®1
s)lim (x³+x²+x+1)/(x²-1) =&nbsp;&nbsp;(3³+3²+3+1)/(3²-1) =&nbsp;&nbsp;(27+9+3+1)/(9-1) =&nbsp;&nbsp;(40)/(8) = 5&nbsp; x®3
t)lim (x²+x)/(x³+x²+x+1) =&nbsp;(0²+0)/(0³+0²+0+1) =&nbsp;(0)/(1) = 0&nbsp; x®0
u)lim 3/(2x+1) =&nbsp;3/(2(1)+1) =&nbsp;3/(3) = 1&nbsp; x®1
v)lim (3x²+x+1)/2 =&nbsp;(3(-1)²+(-1)+1)/2 =&nbsp;(3(1)-1+1)/2 =&nbsp;3/2&nbsp;&nbsp;&nbsp; x®-1
x)lim (x³+2x²+3x+2)/(2x³+x²+2x+4) =&nbsp;&nbsp;(0³+2(0)²+3(0)+2)/(2(0)³+(0)²+2(0)+4) =&nbsp;&nbsp;(2)/(4) = 1/2&nbsp; x®0
2)Calcule os seguintes limites indeterminados:
a)lim (x²-9)/(x-3) = ((x+3)(x-3))/(x+3) = x-3 = 3-3 = 0&nbsp; x®3
b)lim (49-x²)/(7+x) = ((7+x)(7-x))/(x+7) = 7-x = 7-(-7) = 7+7 = 14&nbsp; x®-7
c)lim (5-x)/(25-x²) = (5-x)/((5+x)(5-x)) = 1/(5+x) = 1/(5+5) = 1/10&nbsp; x®5
d)lim (x²+x)/(x²-3x) = (x(x+1))/(x(x-3)) = (x+1)/(1-3) =&nbsp;&nbsp;(0+1)/(0-3) = 1/-3 = -1/3&nbsp; x®0
e)lim x³/(2x²-x) = x³/(x(2x-1)) = x²/(2x-1) = 0²/(2(0)-1) = 0/-1 = 0&nbsp; x®0
f)lim (49+14x+x²)/(7+x) = ((7+x)(7+x))/(7+x) = 7+x = 7+(-7) = 0&nbsp; x®-7
g)lim (x²-6x+9)/(x-3) = ((x-3)(x-3))/(x-3) = x-3 = 3-3 = 0&nbsp; x®3
h)lim (x²-4x+3)/(x-1) = ((x-1)(x-3))/(x-1) = x-3 = 1-3 = -2&nbsp; x®1
i)lim (x²-7x+12)/(x-4) = ((x-3)(x-4))/(x-4) = x-3 = 4-3 = 1&nbsp; x®4
j)lim (x-1)/(x²-3x+2)&nbsp;&nbsp; x®11°, vamos encontrar o produtos notável de&nbsp;(x²-3x+2) &nbsp;encontrando raíz utilizando báskarax²-3x+2 Delta = b²-4acDelta = 9-8Delta = 1
x'=(-b+1)/2ax'=(3+1)/2x'=4/2x'=2
x"=(-b-1)/2ax"=(3-1)/2x"=2/2x"=1
Com : ax²+bx+c = a(x-x')(x-x") então,&nbsp;(x²-3x+2) &nbsp;é igual a 1(x-1)(x-2)
lim (x-1)/(x²-3x+2) &nbsp;= (x-1)/((x-1)(x-2)) = 1/(x-2) = 1/(1-2) = 1/-1 = -1&nbsp; x®1
l)lim (x²-2x+1)/(x-1) =((x-1)(x-1))/(x-1) = x-1 = 1-1 = 0&nbsp; x®1
m)lim (x-2)/(x²-4) =&nbsp;(x-2)/((x-2)(x+2)) =&nbsp;&nbsp;1/(x+2) = 1/(2+2) = 1/4&nbsp; x®2
3) Calcule os seguintes limites indeterminados:
a)lim -x = -(-4) = 4&nbsp; x®-4
b)lim x³= (-4)³ = 16*-4 = -64&nbsp; x®-4
c)lim -x4=&nbsp;-(3)4=&nbsp;-81&nbsp;&nbsp; x®-3
d)lim 3x8&nbsp;=&nbsp;3(-1)8&nbsp;= 3(1) = 3&nbsp; x®-1
e)lim (-x²-3x) = -(5)²-3(5) = -25-15 = -40&nbsp; x®5
f)lim (3x+2)/x² = (3(-2)+2)/(-2)² = (-6+2)/4 = -4/4 = -1&nbsp; x®-2
g)lim (8x+x²) = 8(-1)+(-1)² = -8+1 = -7&nbsp; x®-1
h)lim (5x-x²)/(x-3) =&nbsp;(5(2)-(2)²)/((2)-3) =&nbsp;(10-4)/(2-3) = 6/-1 = -6&nbsp; x®2
i)lim (x²-2x-15) =&nbsp;((-3)²-2(-3)-15) = 9+6-15 = 0&nbsp;&nbsp; x®-3
j)lim (2x³+3x²-7x+3) =&nbsp;(2(-1)³+3(-1)²-7(-1)+3) = 2(-1)+3(1)+7+3 = -2+3+7+3 = -2+13 = 11&nbsp; x®-1
l)lim (4x³-x²-4x²+1) =&nbsp;4(1)³-(1)²-4(1)²+1 = 4-1-4+1 = 0&nbsp; x®1
m)lim (x³-2x²-3x)/(x²-8) =&nbsp;((3)³-2(3)²-3(3))/((3)²-8) = (27-2(9)-9)/(9-8) = (27-18-9)/1=(27-27)&nbsp;/1= 0/1=0&nbsp; x®3
n)lim (4x³+2x²+x+2) = &nbsp;4(-1)³+2(-1)²+(-1)+2 = 4(-1)+2(1)-1+2 = -4+2-1+2 = -5+4 = -1&nbsp; x®-1
o)lim (x4-x³+x²+x+1) = 04-0³+0²+0+1 = 1&nbsp;&nbsp; x®0
p)lim (-2x²-x+2) =&nbsp;-2(3)²-3+2 = -2(9)-1 =-18-1 = -19&nbsp; x®3
q)lim (x³-x²-3)7&nbsp;=&nbsp;((-1)³-(-1)²-3)7&nbsp;=&nbsp;(-1-1-3)7&nbsp;=&nbsp;(-5)7&nbsp;= -25*25*25*5 = -78125&nbsp; x®-1
r)lim (x²+1)4&nbsp;=&nbsp;(2²+1)4&nbsp;=&nbsp;(4+1)4&nbsp;=&nbsp;(5)4&nbsp;=&nbsp;625&nbsp; x®2
s)lim (7x8-x7)/(-2x³+5x²-x+6) =&nbsp;(7(0)8-(0)7)/(-2(0)³+5(0)²-0+6)&nbsp;=0/6 = 0&nbsp; x®0
4) Calcule os seguintes limites indeterminados:
a)lim (x²-16)/(x+4) = ((x-4)(x+4))/(x+4) = x-4 = -4-4 = -8&nbsp; x®-4
b)lim (64-x²)/(8-x) = ((8-x)(8+x))/(8-x) = 8+x = 8+8 = 16&nbsp; x®8
c)lim (9-x)/(81-x²) = (9-x)/((9-x)(9+x)) = 1/(9+x) =1/(9+9) = 1/18&nbsp; x®9
d)lim (3x²+x)/(x²-7x) = (x(3x+1))/(x(x-7)) = (3x+1)/(x-7) =&nbsp;(3(0)+1)/(0-7) = 1/-7 = -1/7&nbsp; x®0
e)lim x5/(5x²-3x) = x5/(x(5x-3)) =&nbsp;x4/(5x-3) = 04/(5(0)-3) = 0/-3 = 0&nbsp; x®0
f)lim (16+8x+x²)/(4+x) = ((4+x)(4+x))/(4+x) = 4+x = 4+(-4) = 0&nbsp; x®-4
g)lim (x²-20x+100)/(x-10) = ((x-10)(x-10))/(x-10) = x-10 = 10-10 = 0&nbsp; x®10
h)lim (x²-9x+20)/(x-5) = ((x-4)(x-5))/(x-5) = x-4 = 5-4 = 1&nbsp; x®5
i)lim (x²+13x+42)/(x+6) = ((x+6)(x+7))/(x+6) = x+7 = -6+7 = 1&nbsp; x®-6
j)lim (x-4)/(x²-7x+12) = (x-4)/((x-4)(x-3)) = 1/(x-3) = 1/(4-3) = 1/1 = 1&nbsp; x®4
l)lim (x²-8x+7)/(x-1) = ((x-1)(x-7))/(x-1) = x-7 = 1-7 = -6&nbsp; x®1
m)lim (x-11)/(x²-121)=(x-11)/((x-11)(x+11)) = 1/(x+11) = 1/(11+11) = 1/22&nbsp; x®11
5) Calcule os limites abaixo:
a)lim (x²+2x-1) =&nbsp;((2)²+2(2)-1) = 4+4-1 = 7&nbsp;&nbsp; x®2
b)lim (y³-2y²+3y-4) =&nbsp;(-1)³-2(-1)²+3(-1)-4&nbsp;= &nbsp;-1-2-3-4 = -10&nbsp; y®-1
c)lim (t²-5)/(2t³+6) =&nbsp;(2²-5)/(2(2)³+6) = (4-5)/(16+6) = -1/22&nbsp; t®2
d)lim (2x+1)/(x²-3x+4) =&nbsp;2(-1)+1)/((-1)²-3(-1)+4&nbsp;= (-2+1)/(1+3+4) = -1/8&nbsp; x®-1
e)lim (y³+8)/(y+2) = ((y+2)(y²-2y+4)) /(y+2) = y²-2y+4 = (-2)²-2(-2)+4 = 4+4+4 = 12&nbsp; y®-2
f)lim (s³-1)/(s-1)= ((s-1)(s²+s+1))/(s-1) = s²+s+1 = 1+1+1 = 3&nbsp; s®1
g)lim (x²+5x+6)/(x²-x-12)=&nbsp;&nbsp; x®-31°, vamos encontrar o produtos notável de&nbsp;( x²+5x+6&nbsp;) e&nbsp;(x²-x-12)&nbsp;encontrando raíz utilizando báskarax²+5x+6 Delta = b²-4acDelta = 25-24Delta = 1
x'= (-b+1)/2ax'= (-5+1)/2x'= -4/2x'= -2
x"= (-b-1)/2ax"= (-5-1)/2x"= -6/2x"= -3
Com : ax²+bx+c = a(x-x')(x-x") então,&nbsp;( x²+5x+6&nbsp;) &nbsp;é igual a 1(x+2)(x+3)
x²-x-12 Delta = b²-4acDelta = 1+48Delta = 49
x'=(-b+7)/2ax'=(1+7)/2x'=8/2x'=4
x"=(-b-1)/2ax"=(1-7)/2x"= -6/2x"= -3
Com : ax²+bx+c = a(x-x')(x-x") então,&nbsp;(&nbsp; x²-x-12&nbsp;&nbsp;) &nbsp;é igual a 1(x-4)(x+3)
lim (x²+5x+6)/(x²-x-12) = ((x+2)(x+3))/((x-4)(x+3)) = (x+2)/(x-4) =&nbsp;(-3+2)/(-3-4) = -1/-7 = 1/7&nbsp; x®-3
h)lim Raíz quadrada de((8r+1)/(r+3)) =&nbsp; h®1Raíz quadrada de((8r+1)/(r+3)) =&nbsp;Raíz quadrada de((8(1)+1)/((1)+3)) =&nbsp;Raíz quadrada de((8+1)/(1+3) =Raíz quadrada de(9/4) =Raíz quadrada de(3²/2²) =3/2
i)lim Raíz quadrada de((x²+3x+4)/(x³+1)) =&nbsp;&nbsp; h®2Raíz quadrada de((x²+3x+4)/(x³+1)) =&nbsp; Raíz quadrada de(((2)²+3(2)+4)/(2³+1)) =&nbsp; Raíz quadrada de((4+6+4)/(8+1)) =&nbsp; Raíz quadrada de(14/9) =&nbsp; (Raíz quadrada de14)/(Raíz quadrada de9) =&nbsp;(Raíz quadrada de14)/3&nbsp;
Resposta:&nbsp;(Raíz quadrada de14)/3&nbsp;
j)lim&nbsp;Raíz quadrada de(((y²-9)/(2y²+7y+3))=&nbsp; y®-3
2y²+7y+3Delta = b²-4acDelta = 49-24Delta = 25
x'=(-b+5)/2ax'=(-7+5)/4x'=-2/4x'=-1/2
x"=(-b-5)/2ax"=(-7-5)/4x"= -12/4x"= -3
Com : ax²+bx+c = a(x-x')(x-x") então,&nbsp;(&nbsp;&nbsp;x²-x-12&nbsp;&nbsp;) &nbsp;é igual a 2(x+1/2)(x+3)
Raíz quadrada de(((y²-9)/(2y²+7y+3))= Raíz quadrada de((y+3)(y-3)/ 2(y+1/2)(y+3))=Raíz quadrada de ((y-3)/&nbsp;2(y+1/2))&nbsp;= Raíz quadrada de ((-3-3)/&nbsp;2(-3+1/2))&nbsp;= Raíz quadrada de ((-6)/&nbsp;2(-6/2+1/2))&nbsp;= Raíz quadrada de ((-6)/&nbsp;2(-5/2))&nbsp;= Raíz quadrada de&nbsp;(-6/-5)&nbsp;= Raíz quadrada de&nbsp;(6/5)&nbsp;=? (Não sei fazer.)
k)lim&nbsp;Raíz quadrada de(((8t³-27)/(4t²-9))=&nbsp; t®3/2
l)lim (raiz quadrada de(x+2)-raiz quadrada de 2)/x=&nbsp; x®0
m)lim (2-raiz quadrada de (4-t))/t=&nbsp; t®0
n)lim (raiz cúbica de (h+1)-1)/h=&nbsp; h®0
o)lim (x³-x²-x+10)/(x²+3x+2)=&nbsp; x®-2
p)lim (2x³-5x²-2x-3)/(4x³-13x²+4x-3)=&nbsp; y®3