ADMINISTRAÇAO PUBLICA

QUESTÕES:
1) Calcule o limite das funções
a)

( x + 1)( x − 1)
= x +1 = 1+1 = 2 x →1
( x − 1)

lim

lim x →1

x² − 1
=2
x −1

b)

( x − 1)( x + 1)
( x − 1)
= lim x →1 ( x + 1)( x − 1)( x + 1) x →1 ( x + 1)( x − 1)( x + 1)
1
1
1
1 lim = lim
= lim
=
x →1 ( x + 1)( x + 1) x →1 (1 + 1)( 1 + 1) x →1 ( 2).( 2)
4

lim

lim x →1

c)

x −1 1
=
x² − 1 4

( x + 1)( x − 1) lim x + 1 = −1 + 1 = 0 x → −1 x →1
( x − 1) x² − 1 lim =0 x → −1 x − 1 lim d)

x2 − 4 lim 2
=
x→2 x − 2x
( x + 2)( x − 2) x+2 2+2 4 lim = lim
=
= =2 x→2 x→2 x ( x − 2) x 2
2
x² − 4 lim =2 x →2 x² − 2 x
2) Verifique se as funções são contínuas em x=c.
a) f ( x ) =

x2 − x − 2
, c=0 x−2 ( x + 1)( x − 2) x² − x − 2
= lim
= lim x + 1 = 0 + 1 = 1 x →0 x→0 x→0 x−2 x−2 x² − x − 2 lim =1 x →0 x−2 lim

x² − x − 2 = 0
∆ = b ² − 4ac
∆ = 1+ 8 = 9 x= x² − x − 2 f ( x) = x−2 0² − 0 − 2 − 2
=
=1
0−2
−2 x² − x − 2 lim = f (0), x →0 x−2 f (0) =

Logo, f(x) é continua para c=0

b) f ( x ) =

( x + 1)( x − 2) x² − x − 2
= lim
= lim x + 1 = 2 + 1 = 3 x→2 x→2 x→2 x−2 x−2 x² − x − 2 lim =3 x→2 x−2

− ( −1) ± 9
2( −1)
1± 3 x= 2
1− 3
= −1 x' =
2
1+ 3 x" =
=2
2 x= x² − x − 2 = 0
∆ = b ² − 4ac
∆ = 1+ 8 = 9 x= x² − x − 2 f ( x) = x−2 2² − 2 − 2 4 − 2 − 2 f ( 2) =
=
=0
2−2
0 f ( 2) = ∃

x→2

x² − x − 2
≠ f (2) logo, a função não é continua em c=2 x−2 3) Calcule a derivada das funções abaixo:
a)

2a

x2 − x − 2
, para c=2 x−2 lim

lim

−b± ∆

−b± ∆
2a

− ( −1) ± 9
2( −1)
1± 3 x= 2
1− 3 x' =
= −1
2
1+ 3 x" =
=2
2 x= d 5 x = 5 x 5−1 = 5 x 4 dx b)

d
5 x ³ = 5.3 x 3−1 = 15 x ² dx c)

d
4 x ³ + 3x ² − x + 5 = 4.3x 3−1 + 3.2 x 2 −1 − x 1−1 + 0 = 12 x ² + 6 x − 1 dx 4) Determinada Empresa de nome fantasia DELTA LTDA produz um determinado produto, com um custo mensal dado pela função:
1
C ( x) = x 3 − 2 x