Page 4 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 1 Limits and Continuity

1.1 Limits

1 Limit Introduction
to Limit
Limits INFO VIDEO
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Let f be a function defined on an open interval containing a, possibly not defined at a.

The limit (or limiting value) of f(x) as x approaches a, written as lim f(x), is the value that f(x) approaches as
x → a
x approaches a. If the limit of f(x) as x approaches a is l, then we write
lim f(x) = l
x → a
or f(x) → l as x → a

1. If f(x) → l as x → a from the left (i.e. values of x less than a), we write x → a – f(x) = l and this is known
lim
as the left-hand limit.

2. If f(x) → l as x → a from the right (i.e. values of x more than a), we write x → a + f(x) = l and this is
lim
known as the right-hand limit.

3. Take note that x → a – f(x) may not be the same as x → a + f(x).
lim
lim
lim
lim
lim
Hence, x → a f(x) = l if and only if x → a + f(x) = x → a – f(x) = l.
Properties of limits

Let a be any real number and k any constant.

1. lim c = c, c is a constant
x → a
2. lim k f(x) = k lim f(x)
x → a x → a

3. 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

4. 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
5. 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

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01 STPM Math(T) T2.indd 2 28/01/2022 5:30 PM