MATH 3751 Code 2256 Real Analysis: Exam 2: In Class Portion Fall 2005
Name: Score: /65
Good Luck!!
1. (15 points) Define the following:
(a) lim
x→c+f(x) = L.
(b) lim
x→c
f(x) = −∞.
(c) lim
x→∞
f(x) = L.
2. (10 points) Let f , g :A→Rand suppose that cis a cluster point of A. If lim
x→c
f(x) = Land
lim
x→c
g(x) = M, then using the ²−δdefinition of limit prove that
lim
x→c
(f(x) + g(x)) = L+M.
3. (10 points) Let f , g :A→Rand suppose that cis a cluster point of A. If lim
x→c
f(x) = Land
lim
x→c
g(x) = M, and if g(x)6= 0 for all x∈A,M6= 0 then using the sequential criterion for
limits prove that
lim
x→c
f(x)
g(x)=L
M.
4. (10 points) Using the ²−δdefinition of limit, prove that
lim
x→−2
x+ 3
(x)(x−1) =1
6.
5. (10 points) Prove that if a sequence of real numbers (xn) is a Cauchy sequence then it is a
convergent sequence.
6. (10 points) Show that if xn≥0 for all nand lim
n→∞
xn= 0 then lim
n→∞
√xn= 0.
