Lecture Notes 2: Review of Limits.
Squeeze Theorem
Instructor: Anatoliy Swishchuk
Department of Mathematics & Statistics
University of Calgary, Calgary, AB, Canada
MATH 265 ’University Calculus I’
L01 Winter 2017
Outline of Lecture
1. Review of Limits
2. Computing Limits
3. Monotone Limits
4. Squeeze Theorem
The Limit of a Function
We write limx→a f (x) = L if we can make the value of f (x)
arbitrary close to L by taking x to be sufficiently close to a (on
either side of a) but not equal to a.
We read it as ’the limit of f (x) as x approaches a is equal to L’.
One-sided Limits
We write limx→a− f (x) = L if we can make the value of f (x)
arbitrary close to L by taking x to be sufficiently close to a and
less than a. We call it left-hand limit.
We write limx→a+ f (x) = L if we can make the value of f (x)
arbitrary close to L by taking x to be sufficiently close to a and
x greater than a. We call it right-hand limit.
We note, that limx→a f (x) = L if and only if
limx→a− f (x) = L and limx→a+ f (x) = L
Computing Limits
There are two general approaches to compute the limits:
1) Graphically and 2) Algebraically:
1. Graphically-using graphs: E.g., for the piecewise function 1.
in Lecture Notes 1, the limit from the left, limx→1− f (x) = 0, is
0, but the limit from the right, limx→1+ f (x) = 1, is 1, meaning
that limx→1 f (x) does not exist.
2. Algebraically: using the limit properties (see next slide) and
combinations of functions f and g such as
(f ± g)(x) = f (x) ± g(x),
(f g)(x) = f (x)g(x),
and composite function (f ◦ g)(x) = f (g(x)).
f
f (x)
( )(x) =
g
g(x)
Monotone Limits
If f (x) ≤ g(x) when x is near a (except possibly at a) and the
limit of f and g both exist as x approaches a, then
lim f (x) ≤ lim g(x).
x→a
x→a
Squeeze Theorem
Suppose that g(x) ≤ f (x) ≤ h(x) for all x close to a but not equal
to a.
If limx→a g(x) = L = limx→a h(x),
then limx→a f (x) = L.
Squeeze Theorem: Examples
Example 1. If for all x ≥ 0, 4x − 9 ≤ f (x) ≤ x2 − 4x + 7, then
limx→4 f (x) = 7, because 4 × 4 − 9 ≤ f (x) ≤ 4 × 4 − 4 × 4 + 7 and
7 ≤ f (x) ≤ 7. Therefore, limx→4 f (x) = 7. We used monotone
limit property and squeeze theorem here.
x
Example 2. limx→0 sin
x = 1 (will be proved in class).
References
1) Calculus: Early Transcendental, 2016, An Open Text, by
David Guichard: https : //lalg1.lyryx.com/textbooks/CALCU LU S
1/ucalgary/winter2016/math265/Guichard
− Calculus − EarlyT rans − U of Calgary − M AT H265 − W 16.pdf
2) Optional Textbook: Essential Calculus, Early Transcendental,
2013, by J. Stewart, 2nd edition, Brooks/Cole