Page 15 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 1 Limits and Continuity

Exercise 1.3

1. Use the intermediate value theorem to show that there is a root of the given equation in the given interval. 1
3
(a) x – 4x + 1 = 0 ; (0, 1)
5
4
(b) x – 2x – x – 4 = 0 ; (2, 3)
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3
2
(c) x + 3x = x + 1 ; (0, 1)
2
(d) x = x + 2 ; (1, 2)
3
2. Show that the equation x + x – 5 = 0 has at least one root a where 1 , a , 2.
–x
3. Show that the equation e + 2 = x has at least one real root.
x
4
4. Use the intermediate value theorem to show that the equation x = 2 has at least one root.
3
5. Use the intermediate value theorem to show that the equation x + x + 1 = 0 has a root in the interval
(–2, 0).

Summary

lim
lim
lim
1. x → a f(x) exists if and only if x → a – f(x) = x → a + f(x).
2. Properties of limits
Let a be any real number and k any constant.

(a) lim c = c, c is a constant
x → a
(b) lim k f(x) = k lim f(x)
x → a x → a
(c) The limit of a sum (or difference) is the sum (or difference) of the limits.
lim [f(x) ± g(x)] = lim f(x) ± lim g(x)
x → a x → a x → a

(d) The limit of a product is the product of the limits.
.
lim [f(x) g(x)] = lim f(x) . lim g(x)
x → a x → a x → a
(e) The limit of a quotient is the quotient of the limits
lim
lim f(x) = x → a f(x) , provided lim g(x) ≠ 0
x → a g(x) lim g(x) x → a
x → a
lim
3. A function f is continuous at a if x → a f(x) = f(a).
4. A function f is continuous on an open interval (a, b) if it is continuous at every number in the interval;
a function f is continuous on a closed interval [a, b] if it is continuous on (a, b) and is also continuous
from the right at a and from the left at b.

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01 STPM Math(T) T2.indd 13 28/01/2022 5:30 PM