Understanding the Foundations of Calculus
Introduction to
Limits and
Continuity
Asst. Prof. Michael A. Manalo
What is a Limit?
Intuitive Definition:
Approaching a Value
● A value a function approaches as input gets
close to a point.
● It describes the function's behavior near a specific
input
When Does a Limit Exist?
The left-hand limit must exist.
The right-hand limit must exist.
Both one-sided limits must agree (LHL = RHL).
If any condition fails, the limit does not exist (DNE).
● Not necessarily the function’s value at that exact
point.
One-Sided Limits
Notation of a Limit
limx→c f(x) = L
Left-Hand Limit: Approaching from values less
than c: lim x→c⁻ f(x)
Right-Hand Limit: Approaching from values
greater than c: lim x→c⁺ f(x)
Evaluating Limits: Techniques and Examples
When Direct Substitution Fails
Indeterminate Forms
➤ Direct substitution is initial
method.
➤ Often yields forms like 0/0 or
∞/∞.
A Strategic Toolkit for Limits
Beyond basic substitution, a diverse array of algebraic and
analytical methods are essential for precise limit evaluation.
➤ These require specific
strategies.
1. Direct Substitution
2. Factoring & Simplifying
Initial Check
Indeterminate 0/0
➤ Applicable for continuous functions.
➤ Factor numerator & denominator.
➤ Evaluate f(c) directly.
➤ Cancel common terms.
➤ Yields the limit if defined.
➤ Re-evaluate after simplification.
3. Rationalizing Expressions
4. Limits at Infinity
Properties of Limits
Simplifying Complex Limit Evaluations
Sum & Difference Rule
Product & Quotient Rule
lim [f(x) ± g(x)] = L ± M
lim [f(x) * g(x)] = L * M
The limit of a sum or difference is the sum or difference of the
individual limits.
lim [f(x) / g(x)] = L / M
*Condition: Limits L & M exist
The limit of a product or quotient is the product or quotient of the
limits.
*Condition: Limits L & M exist (M ≠ 0 for quotient)
Constant Multiple Rule
Power Rule
lim [c * f(x)] = c * L
A constant factor can be moved outside the limit expression.
*Condition: Limit L exists
lim [f(x)]n = Ln
The limit of a function raised to a power equals the limit raised to
that power.
*Condition: Limit L exists, n is a real number
Introduction to Continuity
Understanding the Smoothness of Functions
INTUITIVE DEFINITION
CLASSIFICATION
A function f(x) is continuous at a point x = a if its
When a function fails one or more continuity
graph can be traced without lifting your pencil.
conditions, it results in a discontinuity, or a break in
the graph.
Formal Conditions for Continuity
Types of Discontinuities
f(a) is defined (the point exists).
The limit of f(x) as x approaches a exists (limx→a
f(x)).
The limit equals the function value (lim x→a f(x)
= f(a)).
Removable
A "hole" where the
limit exists but
differs from f(a).
Jump
Left and righthand limits exist
but are not equal,
Infinite
One or both onesided limits
approach ±∞ (a
Testing for Continuity: Examples and Solutions
Checking the Three Conditions of
Continuity
Examples of Discontinuous Functions
Discontinuities occur when any of the three continuity conditions
fail. They are classified into distinct types.
Function must be defined at the point f(a).
The limit of the function must exist at the point
f(x).
lim x→a
The limit must equal the function's value: lim x→a f(x) =
f(a).
Essential Calculus
Foundational Rule
Examples of Continuous Functions
A function is continuous if its graph can be drawn without lifting
your pencil. Polynomials and rational functions are continuous
within their domains.
Removable (Hole)
Jump (Gap)
Infinite (Asymptote)
"If a function is continuous, its limit equals its function value."
Polynomials (e.g., f(x) = x²)
Rational Functions (in domain)
How to Identify Discontinuities
Removable: Limit exists but ≠ f(a) (hole in graph).
Summary: Limits
1. Direct Substitution
Plug in the value directly.
If result is a real number, that's the limit.
Applicable for continuous functions at the
point.
2. Algebraic Factoring
Used for indeterminate forms (e.g., 0/0).
Factor expressions to simplify.
Cancel common factors, then substitute.
3. From a Graph
4. Limits at Infinity
Observe function behavior from left and
right.
Examine function's end behavior (x → ±∞).
Look for horizontal asymptotes.
Summary: Continuity
Problem 1: Point
Continuity Test
Check if a function is continuous at
a point `a` by verifying three
conditions:
Problem 2:
Classify
Discontinuities
Find values that make a piecewise
function continuous.
Is `f(a)` defined?
Locate the boundary points
where the function definition
changes.
Does `lim x→a f(x)` exist?
Is `lim x→a f(x) = f(a)`?
Removable (Hole): Single point missing
or misplaced.
Jump (Gap): Limits from left and right
sides differ.
Infinite (Asymptote): Function value
approaches ±∞.
f(a) Defined
Limit Exists
Problem 3:
Piecewise
Function
Removable
Jump
Set the limits from the left and
right equal to each other at
each boundary.
Solve the resulting equations to
find the unknown values (e.g., c,
k).
Boundary Check
Journey's End: Your Mastery of Limits & Continuity
Recap & Open Discussion
Key Concepts: Limits
Limit Definition: Approaching a value
Techniques for Evaluation:
Key Concepts: Continuity
Continuity Test: 3 Essential Conditions
`f(a)` is defined.
Factoring & Simplifying
Limit as `x`→`a` exists.
Rationalizing (Conjugates)
`f(a)` equals the limit.
L'Hôpital's Rule
Limits at Infinity: End behavior, horizontal asymptotes
Types of Discontinuities:
Removable (Hole)
"A limit is the value a function approaches as the input
approaches a specific point."
Source: BYJU'S - Limits Definition
•
Jump
•
Infinite
"A function is continuous if its graph can be drawn without lifting
your pencil."
Source: Calcworkshop - Limits And Continuity