THE UNIVERSITY OF ZAMBIA

Department of Mathematics and Statistics

MAT1110: Foundation Mathematics and Statistics for Social Sciences
Tutorial Sheet 8 (2020/2021)
1. For each of the following functions, determine whether the limit exists or does not exist
at a given value of x.
(a) (c)
(
if x ≥ 0, x 2 − 16
f(x) := at x = 0 f(x) := , x 6= 4, at x = 4
1
−1 if x > 0, x−4

(b) (d)
( (
7x − 2 if x ≥ 2, x2 + 5 if x < −2,
f(x) := at x = 2 f(x) := at x = −2
3x + 5 if x < 2, 3 − 3x if x ≥ −2,

2. Evaluate each of the following:

(a) i. v. r
x 2 − 3x + 2
 
− 2 .
1 1
x2 − 4
lim
x x
lim
xÏ0 xÏ2

vi.
x3 + x2 − x − 1
ii.
x 2
x−1
lim
xÏ1
xÏ−5 3x − 4
lim
vii.
iii.
x −9 2
x 4 + 3x 3 − 13x 2 − 27x + 36
xÏ3 x − 3
lim
x 2 + 3x − 4
lim
xÏ1

√ √
iv.
x3 − 8
viii.
x+3− 3
xÏ2 x − 2
lim
x
lim
xÏ0

(b) Evaluate each of the following:

i. iv. vii. √
3x − 4x + 2 x2 + 5
lim 2x −5x +3x +1 .
11 6 2 3

3x 2 − 2
xÏ+∞
lim
lim xÏ+∞
xÏ+∞ 7x 3 + 5
ii. v.
lim 3x 4 − x 2 + x − 7 4x 5 − 1
xÏ−∞ lim
xÏ+∞ 3x 3 + 7
iii. vi.
4x − 1
2x + 5 lim √ .
xÏ+∞ x − 7x + 3 xÏ+∞ x2 + 2
lim 2

3. (a) Discuss the continuity of each of the following functions at a given value of x.
i. (
x 2 if x ≤ 0,
f(x) := at x = 0
x if x > 0,
ii. (
if x ≥ 0,
f(x) := at x = 0
1
−1 if x > 0,
 iii. n
x 2 −4
f(x) := x+2
if x 6= −2 at x = −2
iv. 
x + 1
 if x ≥ 2,
f(x) := 2x − 1 if 1 < x < 2, at x = 1 and x = 2

x − 1 if x ≤ 1,
4. (a) Find the derivative of each of the following from the first principles:

i. iv.
y=5 y=
1
x+2
ii. v. √
y = 2x y = x+1
vi.
y=√
iii. 1
y = 2x + 3x + 6
2
x+1

(b) Differentiate each of the following with respect to x :

i. iv.
y=x+
1
x 3x 7 + 5x 5 − 2x 4 + x − 3
y=
ii. x4
y = x2 − √
8
x
v.
√ x 2 + 2x
iii. y = 2x 3 + x+
y = (x + 2)2 x2

(c) i. Show that

f 0 (x) =
3 − 21
x (4 − x),
2
if
f(x) = 12x 2 − x 2
1 3

ii. Expand the right hand side of

f(x) = (x 2 − 1)(x − 2 + 1),

3 1

and hence find f 0 (x).

(d) Difference with respect to x :

i. v. ix.
f(x) = (3x 4 − x 2 )5 f(x) = 4 3x
f(x) = sin(x 2 + 2)
vi.
ii. √
f(x) = x−2 f(x) = ln(2x 2 + 1) x.
iii.
√
vii. f(x) = tan(2x + 1)
f(x) = x 2
x−2 f(x) = log2 (5x + 1)
iv. viii.
f(x) = e3x f(x) = cos(2x)
2