课程名称︰微积分甲上
课程性质︰大一必修
课程教师︰蔡雅如
开课学院：电资院、管理院、社科院
开课系所︰资工系、资管系、经济系
考试日期（年月日）︰2014/10/06
考试时限（分钟）：1HR
试题 :
1.Find the limits (Please show your work):
x^⅓-1
(a) lim ─────
x→1 √(x^3) -1
１
(b) lim sin(─)‧arctan([x])
x→∞ ｘ
１
(c) lim e^(───)
x→0+ x^2 -x
(d) lim √x^2+ax + √x^2+b+2x , where a,b are constants.
x→-∞
＿＿
1-√1-x
2. Let f(x) = {a(────) + [x] , for x≠0.
| |x|
{ b , for x＝0.
(a) Find values of a and b such that f is continuous at x=0.
f(x)－f(0) f(x)－f(0)
(b) For the a, b found in part (a), compute lim ───── & lim ─────
x→0+ ｘ x→0- ｘ
Is f differentiable at x=0?
3.Let f(x)=tan(arccos x).
(a) Simplify f(x) and state the domain and the range of f(x).
(b) Show that the equation f(x) = x has a root insite the interval (½,１).