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Í× 2009 · ¹Á ŠΡ(A) ÁÊ
1. (7 × 6′ = 42 ) ³ÇÏ É (1) lim
n→∞
(
1
√
n2 + 1
+
1
√
n2 + 2
+ . . . +
1
√
n2 + n
);
(2) lim
x→a
sin2
x − sin2
a
x − a
; (3) lim
x→∞
(1 −
2
x
)−x
;
(4) Ø ²Ä Ĥ x +
x3
3
+
x5
5
+ · · · +
x2n+1
2n + 1
+ · · ·;
(5) ³Ç «° +∞
0
e−ax − e−bx
x
dx, Õb > a > 0;
(6) ³ÇI =
L
xdy − ydx
x2 + y2
, ÕL Ì ª¦ § ¨¶ ¥
(7) F(x, y) =
xy
x
y
(x − yz)f(z)dz, Fxy.
2. (8 ) ı ε − δ
Ñ ¤
(1) lim
x→1
x2 − 1
2x2 − x − 1
=
2
3
; (2) lim
(x,y)→(0,0)
x2y
x2 + y2
= 0.
3. (10 ) ℄à ı lim
x→+∞
f(x) µ ¢© ÈÑ lim
x→+∞
cos x ª
£
4. (10 ) Äf(x) § ´[a, b] ¾ ( ¶), ¶ »Ñ f(x)
[a, b] Ù Ó£
5. (10 ) f(x) =
x2, x ≥ 3,
ax + b, x < 3.
a, b Ӣ f(x) x = 3 ƺ
£
6. (10 ) Äf(x) [a, b] º £Ñ ¤ ξ ∈ (a, b),
2ξ[f(b) − f(a)] = (b2
− a2
)f′
(ξ).
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