A limit describes what value a function approaches as the input approaches some value. Limits are the
foundation of both derivatives and integrals - all of calculus is built on this concept.
lim(x→a) f(x) = L
"As x gets closer to a, f(x) gets closer to L"The limit
can exist even if f(a) is undefinedExample: lim(x→0) sin(x)/x = 1
Key limit laws: Limits
distribute over +, −, ×, ÷ (when denominator ≠ 0). L'Hôpital's Rule: if lim gives 0/0 or ∞/∞,
differentiate top and bottom separately.
Worked Example 1
Evaluating a Limit - Factoring Method
Problem: Find lim(x→3) (x² − 9)/(x − 3)
Factor and simplify
Direct substitution: (9−9)/(3−3) = 0/0 → indeterminate
Factor: (x²−9)/(x−3) = (x+3)(x−3)/(x−3) = x+3
lim(x→3) (x+3) = 6
Answer: 6. The function is undefined at x=3, but the limit exists
because we care about behavior as x approaches 3, not at 3 itself. This distinction is the entire reason
limits exist.
2 Derivatives - Rate of Change
The derivative measures the instantaneous rate of change of a function - the slope of the tangent line at
any point. It's defined as the limit of the difference quotient.
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
f'(x) = dy/dx = slope at point xPower rule: d/dx [xⁿ] =
nxⁿ⁻¹Constant rule: d/dx [c] = 0
d/dx [xⁿ]
nxⁿ⁻¹
d/dx [sin x]
cos x
d/dx [cos x]
−sin x
d/dx [eˣ]
eˣ
d/dx [ln x]
1/x
Worked Example 2
Finding Velocity from Position
Problem: A ball's height is h(t) = 50t − 5t² meters. Find its
velocity at t = 3 seconds. When does it reach maximum height?
Differentiate
v(t) = h'(t) = 50 − 10t
v(3) = 50 − 30
v(3) = 20 m/s (still going up)
Max height when v = 0: 50 − 10t = 0
t = 5 seconds, h(5) = 250 − 125 = 125 m
Answer: v(3) = 20 m/s upward; maximum height = 125 m at t = 5 s.
The derivative of position is velocity; the derivative of velocity is acceleration (here: a = −10 m/s²,
i.e., gravity).
3 Chain Rule, Product Rule & Quotient Rule
When functions are combined, special rules apply.
Chain Rule
d/dx [f(g(x))] = f'(g(x)) · g'(x)
Product Rule
d/dx [fg] = f'g + fg'
Quotient Rule
d/dx [f/g] = (f'g − fg')/g²
Worked Example 3
Chain Rule Application
Problem: Differentiate f(x) = (3x² + 1)⁵
Outer function × inner derivative
Let u = 3x² + 1, so f = u⁵
f'= 5u⁴ · du/dx = 5(3x²+1)⁴ · 6x
f'(x) = 30x(3x² + 1)⁴
Answer: 30x(3x² + 1)⁴. The chain rule is essential for any
composite function - "differentiate the outside, multiply by the derivative of the inside."
4 Integration - Reversing Differentiation
Integration is the reverse of differentiation. The indefinite integral finds an antiderivative; the
definite integral calculates the area under a curve.
∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ −1)
C = constant of integration (indefinite)∫ₐᵇ f(x)dx =
F(b) − F(a) (definite)Area under curve between a and b
Worked Example 4
Area Under a Curve
Problem: Find the area under f(x) = x² from x = 0 to x = 3.
Definite integral
∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0
Area = 9 square units
Answer: 9 square units. For comparison, a rectangle 3 wide × 9
tall (max height of x²) has area 27, so the parabolic area is exactly 1/3 of the bounding rectangle - a
result that generalizes to all parabolas.
5 The Fundamental Theorem of Calculus
The theorem that connects differentiation and integration - the most important theorem in all of
calculus. It states that differentiation and integration are inverse operations.
d/dx [∫ₐˣ f(t)dt] = f(x)
Part 1: The derivative of an integral gives back the original
functionPart 2: ∫ₐᵇ f(x)dx = F(b) − F(a) where F' = fThis bridges the
gap between slope and area
6 Applications of Calculus
Calculus appears everywhere in science and engineering.
Derivatives (Rates)
Velocity/acceleration, marginal cost/revenue, population growth rate, reaction rates in
chemistry, current in circuits (dQ/dt).
Integrals (Accumulation)
Distance from velocity, work from force, total charge from current, probability from density
functions, volumes of revolution.
Why calculus matters: Newton
invented calculus to describe planetary motion. Today it underpins all of physics, engineering,
economics, machine learning, and medicine. Every time your phone predicts traffic or a doctor reads
an MRI, calculus is at work.