2004数学四.pdf
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2004
sin x
(1) lim (cos x −b) 5 a = b = .
x
x →0 e −a
14
sin x
lim (cos x −b) 5 lim sin x ⋅(cosx −b) 0
x →0 ex −a x →0
lim (ex −a) 0 a = 1.
x →0
sin x x
lim (cos x −b) lim (cos x −b) 1−b 5 b = −4.
x →0 ex −a x →0 x
a = 1b = −4.
2x
x e dy
(2) y arctan e −ln 2x .
e +1 dx x 1
e −1
e2 +1
x 1 2x
y arctan e −x + ln(e +1)
2
x 2x
e e
′ − +
y 2x 1 2x
1+e e +1
dy e −1
.
2
dx x 1 e +1
⎧ x 2 1 1
xe x
− ≤
,
⎪ 2 2 2
(3) f (x) f (x −1)dx .
⎨ 1 ∫1
⎪−1 , x ≥ 2
⎩ 2
1
−
2
2 1
x − 1 = t ∫1 f (x −1)dx ∫−1 f (t )dt
2 2
1
1 2 1
1 f (x )dt 2 x e x dx + (−1) dx .
∫− ∫− 1 ∫1
2 2
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