This study guide covers the fundamental concepts of functions, limits, and continuity as outlined in the attached document. Understanding these concepts is essential for success in calculus and higher mathematics.
A function from set A to set B is a rule that assigns to each element in A a unique element in B.
Think of a function as a vending machine: you input a specific code (x), and you get a specific snack (f(x)).
The term “function” was recognized by German mathematician Leibniz. Euler used the notation f(x) for functions.
Leibniz → “function”, Euler → f(x) notation
• Area of a square: A = x² (function of side length)
• Volume of a sphere: V = (4/3)πr³ (function of radius)
Functions model real-world relationships between variables.
Key Definition: A function f: A → B is a correspondence that assigns to each element x in A a unique element f(x) in B.
A function f from a set A to a set B is a rule or method or a correspondence that assigns to each element x in A a unique element f(x) in B.
Function Criteria: The correspondence f: A → B is a function if and only if:
If we draw a vertical line parallel to y-axis, if this line intersects a graph in more than one point, it is not the graph of a function.
Imagine dropping a vertical line across a graph. If it touches the graph at more than one point anywhere, it’s NOT a function.
Domain: The set of all possible input values (x-values) for which the function is defined.
Range: The set of all possible output values (y-values) that the function can produce.
Co-domain: The set B in f: A → B (may be larger than the actual range).
Domain = Allowed x-values, Range = Possible y-values, Co-domain = All possible y-values in theory.
Indeterminate: When f(x) takes the form 0/0 for some x = a.
Undefined: When the denominator vanishes (becomes 0) for some x = a.
Functions defined by algebraic expressions. Classified as:
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
Domain: ℝ (all real numbers)
The degree is the highest power of x with non-zero coefficient.
f(x) = ax + b (one-degree polynomial)
Graph is a straight line
Linear functions have constant rate of change (slope).
f(x) = ax² + bx + c (two-degree polynomial)
Graph is a parabola
The vertex of the parabola is at x = -b/(2a).
f(x) = x
Graph is line y = x bisecting I & III quadrants
Identity function: input = output
f(x) = c (fixed value)
Graph is a horizontal line
No matter what x you input, you always get the same output.
f(x) = P(x)/Q(x) where P and Q are polynomials
Domain: All real numbers except where Q(x) = 0
Watch for vertical asymptotes where denominator = 0.
f(x) = aˣ (variable appears as exponent)
Domain: ℝ, Range: (0, ∞)
Exponential growth/decay: a > 1 growth, 0 < a < 1 decay.
f(x) = logₐx
Domain: (0, ∞), Range: ℝ
logₐx = y means aʸ = x. Logarithm is the inverse of exponential.
f(-x) = f(x) for all x
Symmetric about y-axis
Even functions: “E” for “Even” and “E” for symmetry across y-axis.
f(-x) = -f(x) for all x
Symmetric about origin
Odd functions: Origin symmetry (rotate 180°).
sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2
Analogous to trig functions but for hyperbolas
cosh²x – sinh²x = 1 (similar to cos²x + sin²x = 1)
If, as x approaches a from both left and right side, f(x) approaches a specific number L, then L is called the limit of f(x) as x approaches a.
Symbolically: limx→a f(x) = L
Limit Existence Criterion: limx→a f(x) exists if and only if:
limx→a⁻ f(x) = limx→a⁺ f(x) = L
(Left-hand limit = Right-hand limit = L)
If limx→a f(x) = L and limx→a g(x) = M, then:
Limits distribute over addition, subtraction, multiplication, and division (except division by zero).
Forms where a unique value cannot be assigned directly:
When you get 0/0, you can’t conclude the limit doesn’t exist. Try factoring, rationalizing, or L’Hôpital’s Rule.
limx→0 sin(x)/x = 1
limx→0 (1 – cos x)/x = 0
limx→0 tan x/x = 1
sin(x)/x → 1 as x → 0 (x must be in radians!)
limx→0 (eˣ – 1)/x = 1
limx→0 (aˣ – 1)/x = ln a
limx→∞ (1 + 1/x)ˣ = e
e is defined as limn→∞ (1 + 1/n)ⁿ
If limx→a f(x)/g(x) = 0/0 or ∞/∞, then
limx→a f(x)/g(x) = limx→a f'(x)/g'(x)
Only apply L’Hôpital’s Rule to 0/0 or ∞/∞ forms!
A function f is continuous at x = a if and only if:
Simple Definition: A function is continuous at a point if you can draw its graph at that point without lifting your pencil.
limx→a f(x) exists but f(a) ≠ limit or f(a) undefined
Example: f(x) = (x²-1)/(x-1) at x=1
Can “remove” by redefining f(a) = limx→a f(x)
Left-hand limit ≠ Right-hand limit
Example: f(x) = {x for x<0, x+1 for x≥0} at x=0
The graph “jumps” from one value to another at the point.
Function approaches ±∞ at the point
Example: f(x) = 1/x at x=0
Vertical asymptotes indicate infinite discontinuities.
If f and g are continuous at x = a, then:
All polynomial functions are continuous everywhere. All rational functions are continuous except where denominator = 0.
Test your knowledge with these 50 multiple choice questions. Select your answer and check the explanation.
• Understand the definition of a function thoroughly
• Practice domain and range calculations
• Learn to identify functions using the vertical line test
Spend extra time on concepts you find confusing. Don’t rush to advanced topics.
• Draw graphs for different function types
• Use graphing calculators or software
• Sketch limits approaching from left and right
Visualizing helps understand continuity and discontinuity points.
• Solve different types of limit problems
• Work with various function combinations
• Practice both algebraic and graphical approaches
Try solving problems without looking at solutions first.
1. Read each question carefully
2. Identify what’s being asked (domain, range, limit, continuity)
3. Check for special cases (undefined points, asymptotes)
4. Verify your answer makes sense graphically
• Function definitions and notation
• Domain and range calculations
• Function combinations and compositions
• Even and odd functions
• Limit definitions and intuition
• Basic limit calculations
• One-sided limits
• Limit theorems
• Special trigonometric limits
• L’Hôpital’s Rule
• Continuity definitions
• Types of discontinuities
Final Advice: Mathematics is not a spectator sport. You learn by doing. Work through as many problems as possible, and don’t be afraid to make mistakes—that’s how you learn!