Page 17 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 1 Limits and Continuity
7. The function f is defined by

3 – 2e , x , 0
x
f(x) = 2, x = 0 1
3e – 2, x . 0
x
(a) Determine the existence of limit f(x) as x approaches 0.
Penerbitan Pelangi Sdn Bhd. All Rights Reserved.
(b) State, with a reason whether f(x) is continuous at x = 0. Hence, determine the intervals on which f
is continuous.

8. The function f is defined by
f(x) = ln x, 0 , x , 1,
2
ax + b, 1 < x , ∞.
Given f(2) = 3, determine the values of a and b for which f is continuous on (–∞, ∞).

9. The function f is defined by
2
x – 9 , x ≠ 3
f(x) = |x – 3|
6, x = 3
Determine whether f is continuous at x = 3.

10. Evaluate
x – 5
4h
lim
lim
(a) h → 0 e – 1 (b) h → –∞  7
2h
2
4x +
e – 1
11. A function f is defined by
f(x) = ln (x + 2), –2 , x , ∞.
(a) Show that f is continuous on its domain.
(b) Sketch the graph of f.
12. Use the intermediate value theorem to show that there exists a solution to the equation cos x = x in the
interval [0, π ].
2
13. Evaluate
2
2
3x + 16
16x +
lim
lim
(a) x → 0  – 4 (b) x → ∞  1
x
2
2x – 1
14. Show that the graph of y = x – 4x + 1 intersects the x-axis in the interval [0, 2]. Can the same be said
2
for the graph of y = 2x – 3 ?
x – 1
15. The function f is defined by
2
f(x) = x + 2x + 8, x , 0
x
2e + c, x > 0
lim
lim
(a) Find x → 0 – f(x) and x → 0 + f(x). Hence, determine the value of c such that function f is continuous
at x = 0.
(b) Describe the continuity of the function f for
(i) x = 0 (ii) x , 0 (iii) x . 0
15

01 STPM Math(T) T2.indd 15 28/01/2022 5:30 PM