Download Section 3.4: The Sandwich Theorem and Some Trigonometric Limits

Section 3.4: The Sandwich Theorem and Some Trigonometric Limits
Theorem: (Sandwich Theorem)
If f (x) ≤ g(x) ≤ h(x) for all x in an open interval that contains a and
lim f (x) = lim h(x) = L,
x→a
x→a
then
lim g(x) = L.
x→a
Example: If 4x ≤ f (x) ≤ 2x4 − 2x2 + 4 for all x, evaluate lim f (x).
x→1
Example: Show that lim x cos
x→0
1
= 0.
x
sin x
= 0.
x→∞ x
Example: Show that lim
1
Theorem: (Special Trigonometric Limits)
The following trigonometric limits will be used when discussing derivatives.
sin x
=1
x→0 x
lim
1 − cos x
= 0.
x→0
x
and
lim
Example: Evaluate the following limits.
(a) lim
x→0
sin(2x)
3x
sin2 x
x→0
x
(b) lim
1 − cos(2x)
x→0
3x
(c) lim
2
1 − cos(x/2)
x→0
x
(d) lim
sec x − 1
x→0 x sec x
(e) lim
3