TRIG PRECALCULUS HONORS
Section 2.3 THE LIMIT LAWS
The Limit Laws Suppose that c is a constant and the limits
lim f ( x) x→a
1.
lim[ f ( x) + g ( x)] = lim f ( x) + lim g ( x)
2.
lim[ f ( x) − g ( x)] = lim f ( x) − lim g ( x)
x →a
x →a
x →a
x→a
exist. Then,
x →a
3.
lim[cf ( x)] = c lim f ( x)
4.
lim[ f ( x) g ( x)] = lim f ( x) ⋅ lim g ( x)
x →a
x →a
x →a
f ( x) f ( x) lim x→a = lim x→a g ( x ) lim g ( x)
5.
lim g ( x)
x →a
x→a
x →a
and
x →a
if
x →a
[
6.
lim[ f ( x)] = lim f ( x)
7.
lim c = c
8.
lim x = a
9.
lim x n = a n
n
x→a
x→a
lim g ( x) ≠ 0 x →a
]
n
x→a
x →a
x→a
10.
lim n x = n a
11.
lim n f ( x) = n lim f ( x)
x→a
x→a
Rick Villano
(If n is even then a > 0 )
x→a
Trig/Pre-calculus Honors
Page 1
TRIG PRECALCULUS HONORS
Section 2.3 THE LIMIT LAWS
Direct Substitution Property
If f is a polynomial or a rational function and a is in the domain of f, then
lim f ( x) = f (a ) x →a
Theorem 1 (Based on definition of a limit# 3 One sided limits)
lim f ( x) = L x→a
iff
lim f ( x) = L
x→a −
and
lim f ( x) = L
x→a +
Theorem 2 If f ( x) ≤ g ( x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then
lim f ( x) ≤ lim g ( x) x→a
x→a
Theorem 3 The Squeeze Theorem
If f ( x) ≤ g ( x) ≤ h( x) when x is near a (except possibly at a) and
lim f ( x) = lim h( x) = L x→a
Rick Villano
x→a
then,
lim g ( x) = L x→a
Trig/Pre-calculus Honors
Page 2
