課程名稱︰微積分甲下
課程性質︰必修
課程教師︰周青松
開課學院：理學院
開課系所︰地質科學系 生物環境系統工程系 生物產業機電工程系 工管系科管組
考試日期（年月日）︰104/3/13
考試時限（分鐘）：50分鐘
試題 :
Calculus Quiz 1 (104.3.13)
(1) (20 pts)
Show that, if 0<c<d, then the sequence
an = (c^n+d^n)^1/n
is bounded and monotonic.
(2) (20 pts)
Given that
(1+1/n)^n → e, show that
(1+1/n)^(n+1) → e
(3) (20 pts)
State whether the sequence converges as n → ∞;
if it does, find the limit
(a) (1-1/n)^n
(b) (1+π/n)^(3n)
(4) (20 pts)
Given that f is continuous, use L' Hopital rule to determine
lim (1/x ∫x f(t) dt )
x→0 0
(5) (20 pts) Calculate
(a) lim (1/x ∫x e^(t^2) dt )
x→∞ 0
(b) lim (√n - 1 )^1/√n
n→∞