Analysis Problems #2
1. Let f be a function which is defined on the interval [0, 2]. Show that
inf f (x) ≥ inf f (x).
0≤x≤1
0≤x≤2
2. As we are going to prove in class, limits of polynomial functions can be computed by
simple substitution. For instance, one has
lim (2x2 − 4x + 1) = 2 · 32 − 4 · 3 + 1 = 7.
x→3
Assuming this fact for the moment, compute each of the following limits:
2x3 − 5x − 6
,
x→2
x−2
lim
x3 − 3x + 2
.
x→1
(x − 1)2
lim
3. Let f be the function defined by
{
f (x) =
7 − 2x
3x − 3
if x ∈ Q
if x ∈
/Q
}
.
Use the ε-δ definition of limits to show that lim f (x) = 3.
x→2
4. Evaluate the limit lim f (x) when f is the function defined by
x→1
{
f (x) =
1 + 3x
5
if x ̸= 1
if x = 1
5. Use the ε-δ definition of limits to show that lim x2 = 4.
x→2
}
.