Faculty of Management Mathematics Exercises
Sheet 4. Limits and Continuity of Functions
Exercise 4.1. Find the limits (if they exists):
a) lim
x→1(x2+ 5x − 6) b) lim
x→3
x2+ 1
x2− 1 c) lim
x→−∞
x3− 2x2 5x3+ x2− x + 2 d) lim
x→−∞(−2x5+ 6x4− 3x + 7) e) lim
x→0
x + x2
x f) lim
x→1
x3 − x x2− 1 g) lim
x→1
x2+ x − 2
x2− 3x + 2 h) lim
x→12
8x3− 1
6x2− 5x + 1 i) lim
x→1
µ 1
x − 1− 2 x2− 1
¶
j) lim
x→1
(x − 1)√ 2 − x
x2− 1 k) lim
x→4
√1 + 2x − 3
√x − 2 l) lim
x→3
√x + 13 − 2√ x + 1 x2− 9
m) limx→∞¡√
1 + x + x2−√
1 − x + x2¢
n) limx→∞¡√
x + 3 −√ x + 1¢
o) limx→∞x · sinx1
p) lim
x→∞
µx2+ 1 x2− 2
¶x2
q) lim
x→∞
µ2x + 3 2x + 1
¶x+1
r) lim
x→∞
µ3x − 1 2x + 1
¶2x−5
Exercise 4.2. Compute the one-sided limits of the given functions as x approaches x0: a) f (x) = 1
x − 3, x0 = 3 b) f (x) = 1
3 − x, x0 = 3 c) f (x) = 1
(3 − x)2, x0 = 3 d) f (x) = x + 1
x − 1, x0 = 1 e) f (x) = 1
x2− 4, x0 = 2 f) f (x) = 2x−11 , x0 = 1 g) f (x) = 4x2−41 , x0 = 2 h) f (x) = e4−x21 , x0 = −2 i) f (x) = x
1 + ex1, x0 = 0 Exercise 4.3. Finding the one-sided limits verify if the given limits exists:
a) limx→1x + 1
x − 1 b) limx→1|x − 1|3
x3− x2 c) limx→1e1−x21 Exercise 4.4. Decide, if f is a continuous function:
a) f (x) =
( 2x+ 3 if x ≤ 0
(x − 2)2 if x > 0 b) f (x) =
( x − 1 if x < 0 3x if x ≥ 0
c) f (x) =
( e1−xx if x 6= 1
0 if x = 1 d) f (x) =
( sin x
x if x 6= 0 0 if x = 0
e) f (x) =
( x2−1
x+1 if x 6= 1
6 if x = 1 f) f (x) =
( x2− 2 if x 6= −3 5 if x = −3
Last update: November 3, 2008 1
