MATH 421 Lecture Notes - Lecture 1: Horse Length
Problems 6
Please, do problems 2, 4, 5, 7, 11, 12 only.
1. Show that f(x) = x2approaches 4 near x= 2.
2. Show that
lim
x→8√x+ 1 = 3
3. How do you show that
lim
x→0+
1
x=∞
4. Adapt the definition of limit to show:
lim
x→∞
1
x= 0
5. Show that
lim
x→0xsin 1
x= 0
6. Show that for f(x) = x2
lim
x→−∞
f(x) = ∞
7. Given
f(x) = 1x≤0
0x > 0
Show that
lim
x→0f(x)6=L
8. Suppose that the function
f(x) = 0xirrational
1
qx=p
qin lowest terms , p, q ∈N
Let a∈(0,1). Show that
lim
x→af(x) = 0.
9. Assume that limx→cf(x) = Land that limx→cg(x) = M. Find the limit, if there is one and
prove it. The limit could be ∞. Alternatively, state that the limit is indeterminate and give an
example where this happens.
(a) If L= finite and M= finite, what is limx→c
f(x)
g(x)?
(b) If L=∞and M= finite, what is limx→cf(x)g(x)?
(c) If L=∞and M=∞, what is limx→cf(x)−g(x)?
(d) If L=∞and M=∞, what is limx→c
f(x)
g(x)?
10. Prove that if limx→cf(x) = k > 0, limx→cg(x) = 0 and g(x)<0 for all |x−c|< δ for some
δ > 0, then limx→c
f(x)
g(x)=−∞.
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