CSUN Math 150A: Calculus I - Comprehensive Study
Notes
Topic: The Definitions and Properties of Infinitesimals and Infinities
Course: Math 150A (Calculus I)
Institution: California State University, Northridge (CSUN)
Focus: Formal Definitions, Algebraic Properties, and Common Error Analysis
I. Infinitesimals (Quantities Tending to Zero)
The concept of an infinitesimalis perhaps the most fundamental building block for the rigorous
definition of both the derivative and the integral. It is a dynamic concept, not a static number.
A. Formal Definition
A function
f(x)
is called an infinitesimal as
x
approaches a point
a
(where
a
can be a finite
number,
∞
, or
− ∞
) if its limit at that point is zero.
lim
x→ a
f(x)=0
Important Note: The point
a
is vital. A function is only an infinitesimal relative to a specific
limiting process.
Examples:
f(x)=x3
: Infinitesimal as
x → 0
, because
lim
x →0
x3=0
.
g(x)= 1
√
x
: Infinitesimal as
x → ∞
, because
lim
x →∞
1
√
x=0
.
h(x)=sin (x2)
: Infinitesimal as
x → 0
.
B. Personal Insight: The Dynamic Nature
My Insight: Dont think of an infinitesimal as the number
0
. Think of it as a process of
convergence. When we write "Let
Δ x
be an infinitesimal," we are acknowledging that
Δ x
is a
variable quantity not equal to zero but capable of getting closer to zero than any pre-defined
positive number. This non-zero but approaching-zero status is what allows us to divide by it (in
the derivative definition) before the limit is taken. If it were
0
, the algebra would fail.
C. Algebraic Properties of Infinitesimals
If
α(x)
and
β(x)
are infinitesimals as
x → a
, then:
Sum and Difference: The sum or difference of two infinitesimals is still an infinitesimal.
lim
x→ a
(α(x)± β(x))=lim
x→ a
α(x)±lim
x→ a
β(x)=0±0=0
.
Product: The product of two infinitesimals is an infinitesimal.
lim
x→ a
(α(x)⋅β(x))=(lim
x→ a
α(x))⋅(lim
x →a
β(x))=0⋅0=0
.
Constant Multiple: The product of an infinitesimal and a constant
c
is an infinitesimal.
lim
x→ a
(c⋅α(x))=c⋅lim
x → a
α(x)=c⋅0=0
.
Product with a Bounded Function (The Squeeze Theorem Helper):
If
α(x)
is an infinitesimal as
x → a
, and
g(x)
is a bounded function (meaning there exists
M>0
such that
¿g(x)∨≤ M
for
x
near
a
), then the product
α(x)⋅g(x)
is an infinitesimal.
Practical Example: We use this to solve limits like
lim
x →∞
sin(x)
x
. Here,
α(x)= 1
x
(infinitesimal) and
g(x)=sin (x)
(bounded by
M=1
). Since
infinitesimal ×bounded=infinitesimal
, the limit
is
0
.
II. Infinities (Quantities Tending to Unbounded Growth)
An infinity is the counterpart to an infinitesimal, representing unbounded growth.
A. Formal Definition
A function
f(x)
is called an infinity as
x
approaches
a
if the limit of its absolute value grows
without bound (i.e., tends to positive infinity).
lim
x→ a
¿f(x)∨¿∞
This is usually abbreviated in notation to express the direction of growth:
lim
x→ a
f(x)=∞or lim
x → a
f(x)=−∞
Examples:
f(x)= 1
¿¿
: Infinity as
x → 2
, because
lim
x→ 2
1
¿¿
.
g(x)=ex
: Infinity as
x → ∞
.
h(x)=− x4
: Infinity as
x → ∞
, specifically
lim
x→ ∞
h(x)=− ∞
.
B. The Critical Reciprocal Relationship
This is the most powerful connection between the two concepts and the one we use constantly
for simplifying limits.
If
f(x)
is an infinitesimal, then its reciprocal
1
f(x)
is an infinity.
If
lim
x→ a
f(x)=0
, then
lim
x → a
1
f(x)=∞
(or
− ∞
, depending on the sign of
f(x)
near
a
).
If
g(x)
is an infinity, then its reciprocal
1
g(x)
is an infinitesimal.
If
lim
x→ a
g(x)=± ∞
, then
lim
x → a
1
g(x)=0
.
Practical Application: When we evaluate
lim
x →∞
3x2+1
x2−5x
, we divide every term by
x2
. The result is
3+1
x2
over
1−5
x
. The terms
1
x2
and
5
x
immediately become infinitesimals (they tend to 0),
simplifying the limit evaluation to
3/1=3
.
III. Indeterminate Forms: The Algebraic Challenge
An Indeterminate Form arises when an algebraic operation involves infinitesimals and/or
infinities, but the result cannot be determined without further algebraic manipulation. It signifies
a "tie" between two competing processes (one tending to zero, the other tending to infinity).
The Seven Indeterminate Forms (Focus on the First Four for Calc I)
Form
Description
Practical Resolution in Math 150A
0
0
Quotient of two infinitesimals
Factoring, Rationalizing, Simplification (Algebra)
∞
∞
Quotient of two infinities
Dividing by the highest power of
x
(Algebra)
0⋅∞
Product of an infinitesimal and an infinity
Convert to
0
0
or
∞
∞
using the Reciprocal Rule (e.g.,
f⋅g=f
1/g
)
∞ −∞
Difference of two infinities
Rationalizing expressions (especially with
√
❑
), Finding a common denominator, or Factoring out
the dominant term
1∞
Exponential
Requires using logarithms (LHôpitals Rule prep)
00
Exponential
Requires using logarithms (LHôpitals Rule prep)
∞0
Exponential
Requires using logarithms (LHôpitals Rule prep)
My Insight: Always remember the goal in resolving these forms is to use algebra to turn them into
a DETERMINATE form, like
c
0
,
0
c
,
∞
c
, or
c⋅∞
, where
c
is a non-zero constant. Only then can you
evaluate the limit.
IV. Common Pitfalls and Error Analysis
These are the typical traps that catch students in Math 150A when dealing with these concepts.
Pitfall 1: Assuming
0
0
or
∞
∞
Always Result in 1 or 0
The Error: Many students incorrectly assume that if the numerator and denominator are both
approaching zero, they "cancel out" to 1, or that if the numerator is 0, the result must be 0.
The Reality: The result of
0
0
or
∞
∞
depends on the rate of approach.
If the numerator approaches zero slower than the denominator (e.g.,
lim
x → 0
x
x2
), the result is
∞
.
If the numerator approaches zero faster than the denominator (e.g.,
lim
x → 0
x2
x
), the result is
0
.
If they approach zero at the same rate (e.g.,
lim
x → 0
5x
x
), the result is a finite, non-zero number (5).
The Fix: You must resolve the algebra until the original factor causing the indeterminate form is
canceled out. For example, in
lim
x→ 2
x2−4
x −2
, the
x − 2
factor must be eliminated by factoring the
numerator.
Pitfall 2: Misinterpreting
∞ −∞
The Error: Assuming
lim
x→ ∞
(
√
x2+1− x)=0
because
x2+1
and
x2
are nearly the same for large
x
,
implying the infinities cancel out perfectly.
The Reality:
∞ −∞
can be any value (
c
,
∞
,
− ∞
, or
0
). The difference is often small but nonzero,
and it determines the limit.
The Fix (especially for expressions with radicals): When you see
∞ −∞
involving square roots,
you must multiply by the conjugate to transform the expression into the resolvable
∞
∞
form.
√
x2+1− x=(
√
x2+1− x )⋅
√
x2+1+x
√
x2+1+x
=(x2+1)− x2
√
x2+1+x
=1
√
x2+1+x
Now, the limit is
1
∞+∞=1
∞=0
.
Pitfall 3: Failing to Factor Out the Dominant Term for
∞
∞
The Error: For limits at infinity, trying to evaluate the limit term by term before dividing.
The Fix (The Dominance Principle): For rational functions, the limit as
x → ± ∞
is determined
solely by the ratio of the highest-degree terms (the dominant infinities) in the numerator and
denominator.
Growth Hierarchy (from slowest to fastest growth as
x → ∞
):
ln (x)≪xn≪eax(a>0)
Logarithmic Growth:
ln (x)
is the slowest infinity.
Polynomial Growth:
xn
is moderate.
Exponential Growth:
eax
is the fastest infinity.
When dividing infinities, the fastest growing one always "wins."
lim
x →∞
x10
ex=0
(Exponential wins, denominator is dominant).
lim
x →∞
x2
ln (x)=∞
(Polynomial wins, numerator is dominant).
Pitfall 4: Misusing the Bounded Function Property
The Error: Assuming that if
f(x)
is an infinity and
g(x)
is a bounded function, the product
f(x)⋅g(x)
must be an infinity.
The Reality: The product
f(x)⋅g(x)
might be an infinity, but it might also not exist (DNE) if the
bounded function
g(x)
oscillates and does not approach a single value.
Example 1 (DNE):
lim
x→ ∞
x⋅sin(x)
. Here,
x → ∞
and
sin (x)
oscillates between
−1
and
1
. The
function oscillates between
− ∞
and
∞
, so the limit DNE.
Example 2 (Infinity):
lim
x→ ∞
x⋅(sin (x)+2)
. Here,
sin (x)+2
is bounded between
1
and
3
, but is
always positive. The product grows to
∞
.
The Fix: The Product of Infinitesimal and Bounded results in an infinitesimal. The Product of
Infinity and Bounded only results in infinity if the bounded function approaches a non-zero value,
or is bounded away from zero. Otherwise, it DNE.
V. Synthesis and Conclusion
Understanding infinitesimals and infinities is the key to successfully navigating the limit
calculations in Math 150A. These concepts are not abstract numbers but descriptions of function
behavior near a specific point or at the ends of the number line.
When you see
0
in a denominator, its almost always an infinitesimal causing a vertical asymptote
(an infinity).
When you see
x → ± ∞
, you are evaluating the competition between infinities (growth rates).
