Page 8 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 1 Limits and Continuity
8. A function f is defined by
2x, x < 1,
f(x) = x + 1, 1 , x < 2,
1
–x – 1, x . 2.
Evaluate
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(a) lim – f(x), (b) lim + f(x), (c) lim f(x),
x → 1 x → 1 x → 1
(d) lim – f(x), (e) lim + f(x), (f) lim f(x).
x → 2 x → 2 x → 2

1.2 Continuity

Continuity at a number

A function f is said to be continuous at a number a if the graph of f is unbroken at the point (a, f(a)).
The definition implicitly requires that the following three conditions hold:
(a) f(a) is defined,
(b) lim f(x) exists,
x → a
(c) lim f(x) = f(a)
x → a
By definition, function f is continuous at a
if lim f(x) = f(a)
x → a

y y
y = (x)
f(a)

x x
0 a 0 a

Figure 1.1 Figure 1.2
Function f is continuous at x = a Function f is discontinuous at x = a

Example 4

Sketch the graphs of each of the following functions and state whether the function is a continuous function.
4 + x , x , 0 x – 1, x < 0
2
(a) f(x) = (b) f(x) =
4 – x , x > 0 x , x . 0
2

6

01 STPM Math(T) T2.indd 6 28/01/2022 5:30 PM