Calculus (MAT - 1000)
Problems on limit and functions 2
1. Give us following examples:
• A strictly positive function with a zero limit (at a point)
• A function with a removable discontinuity
• A function which has right limit at every point (on its domain) but does not
have left limit everywhere.
• A function which has left limit at every point (on its domain) but does not
have right limit everywhere.
2. Consider the following functions:
(a)
{
1
f (x) =
0
if x ∈ Q
otherwise.
and show that it does not have limit anywhere.
(b)
{
0
f (x) =
1
if x ∈ Q
otherwise.
and show that it has limit only at zero.
3. Consider the following functions and examine them in terms of existence of limits.
If limit exist at all points then mention that. If it fails to exist at some points then
mention that. At these points explore existence of right limit and left limit.
• f (x) = |x|.
•
•
{
x/|x|
f (x) =
0
{
x
f (x) =
0
1
if x ̸= 0
otherwise.
if x ∈ Q
otherwise.
• f (x) = m if x ∈ [m, m + 1) for some m ∈ Z. This is also denoted as f (x) =
⌊x⌋.
• f (x) = x − ⌊x⌋.
• f (x) = m if x ∈ (m, m + 1] for some m ∈ Z. This is also denoted as f (x) =
⌊x⌋.
4. Compute the following limits (limit may not exist, in that case just mention that):
• limx→0
x
|x|
and limx→0
x2
|x|
• limx→0 x sin(1/x) and limx→0 x cos(1/x)
•
√
x−1
x−1
√
x−1
limx→1 x−1
• limx→1
5. Give us examples of functions (defined on R) with the following discontinuity sets
(where the function fails to be continuous):
• {0}
• {1, 2, 3}
• N.
• Z.
• R
6. Comment on continuity of the following functions. If not continuous then mention
the corresponding discontinuity sets clearly:
• f (x) = x − ⌊x⌋.
•
•
•
{
x
f (x) =
2x − 1
{
x2
f (x) =
0
if x ∈ [0, 1)
if x ∈ [1, 2].
if x ∈ [0, 1] ∩ Q
if x ∈ [0, 1] \ Q.


|x|
f (x) = x3 + 5x


3
if x ∈ [−1, 0)
if x ∈ (0, 1]
if x = 0.
Can you make this function continuous by changing value at one point only?
2