2025 Tshwane University Of Technology
Dept. of Mathematics and Statistics
EM115AB - Engineering Mathematics IA
March 3–7, 2025
Worksheet 4
Student #:
Surname & Initials:
1.
Explain in your own words what is meant by
lim f (x) = 5.
x7→2
Is it possible for this statement to be true yet f (2) = 3? Explain fully.
2.
Explain what it means to say that
lim f (x) = 3 and lim+ f (x) = 7.
x7→1−
x7→1
In this situation is it possible that lim f (x) exists?
x7→1
3.
a)
4.
a)
Explain the meaning of each of the following:
lim f (x) = ∞
b) lim+ f (x) = −∞
x→−2
x→4
Consider the graph of f given below, use it to evaluate the whether each quantity exists. If it
does not exist, explain why.
lim f (x)
x→0
b)
lim f (x)
c)
x→3−
lim f (x)
x→3+
e)
f (3)
5.
Consider the graph of R given below, use it to state the following.
1
d)
lim f (x)
x→3
a)
lim R(x)
b)
x→2
lim R(x)
c)
x→5
lim R(x)
d)
x→−3−
lim R(x)
x→−3+
e)
The equations of the vertical asymptotes.
6.
a)
Sketch the graph of an example of a function f that satisfies all of the given conditions:
lim− f (x) = 2, lim+ f (x) = −2 and f (1) = 2
x→1
b)
x→1
lim f (x) = 1, lim+ f (x) = −1, lim− f (x) = 0, lim+ f (x) = 1, f (2) = 1
x→0−
x→0
x→2
x→2
and f (0) is undefined.
c)
lim f (x) = 3, lim− f (x) = 3, lim+ f (x) = −3 , f (1) = 1 and f (4) = −1
x→1
x→4
7.
Evaluate
a)
lim +
x→−3
d)
8.
x→4
x+2
x+3
x+2
x→−3 x + 3
x2 − 2x
lim− 2
x→2 x − 4x + 4
b)
lim cot x
lim −
e)
x→π
lim ln(x2 − 9)
x→3+
Find the vertical asymptotes of the function
y=
9.
c)
x2 + 1
.
3x − 2x2
Given that
lim f (x) = 4 lim g(x) = −2 lim h(x) = 0
x→2
a)
d)
10.
a)
d)
g)
11.
a)
12.
a)
x→2
x→2
find the limits that exist. If the limit does not exist explain why.
lim [f (x) + 5g(x)]
b)
lim [g(x)]3
c)
x→2
x→2
lim g(x)
e)
x→2 h(x)
lim
x→2
p
f (x)
lim g(x)h(x)
f (x)
x→2
Evaluate the limit, if it exists.
x2 + x − 6
lim
x→2
x−2
(4 + h)2 − 16
lim
h→0
h
√
x+2−3
lim
x→7
x−7
x2 − 4x
lim
x→4 x2 − 3x − 4
x+2
lim 3
x→−2 x + 8
1
1
lim
−
t→0
t t2 + t
b)
e)
h)
c)
f)
i)
Use the Squeeze Theorem to show that: √
lim (x2 cos(20πx)) = 0
b)
lim x3 + x2 sin( 20
)=0
x
x→0
x→0
Let
(
4 − x2
f (x) =
x−1
Find lim− f (x) and lim+ f (x).
x→2
x→2
2
if x ≤ 2
if x > 2
x2 − 4x
lim
x→−1 x2 − 3x − 4
√
1+h−1
lim
h→0
h
√
4− x
lim
x→16 16x − x2
b)
Does lim f (x) exist?
c)
Sketch the graph of f .
13.
Evaluate:
14.
√
6−x−2
lim √
x→2
3−x−1
Find the limit if it exist. If the limit does not exist explain why.
a)
d)
x→2
lim (2x + |x − 3|)
x→3
1
1
lim
−
x→0−
x |x|
b)
e)
2x + 12
x→−6 |x + 6|
1
1
lim
−
x→0+
x |x|
lim
c)
2 − |x|
x→−2 2 + x
lim
EXAM–TYPE QUESTIONS
6
15. Find lim 3 + 2 .
x→0
x
(A) 9 (B) 3 (C) ∞ (D) −∞ Answer:
16.
If lim f (x) = m and lim g(x) = n, then lim (xf (x) + g(x)2 ) is equal to
x→3
x→3
x→3
(A) m + n. (B) m + n2 . (C) 3m + n. (D) 3m + n2 . Answer:
17.
Using relevant calculations, determine if the given function is continuous or discontinuous
x3 + 2x − 33 if x ≤ 3,
f (x) = x2 − 6x + 9
if x > 3.
x−3
3
0
