Rucete ✏ AP Calculus In a Nutshell
2. Limits and Continuity
This chapter introduces the concept of limits, different types of discontinuities, and foundational theorems such as the Squeeze Theorem, Intermediate Value Theorem, and Extreme Value Theorem.
Understanding Limits
• A function f(x) has a limit L as x → c if values of f(x) approach L as x approaches c from both sides.
• Notation: – Left-hand limit: limx→c⁻ f(x) – Right-hand limit: limx→c⁺ f(x) – Limit exists only if both are equal
Types of Limits
• One-sided limits: only consider approach from one direction
• Infinite limits: f(x) → ∞ or –∞ near x = c (limit does not exist)
• End behavior: – limx→∞ f(x) – limx→–∞ f(x)
Limit Examples
• Step functions like [x] have different left/right limits at integers → limit DNE at those points
• Absolute value functions: – limx→0 |x|/x = does not exist – limx→0 |x| = 0
• Rational functions: – Asymptotic behavior at points where denominator = 0
Asymptotes
• Vertical asymptote: limx→a⁺ or a⁻ f(x) = ∞ or –∞
• Horizontal asymptote: limx→∞ f(x) = L (a constant)
• Unlike vertical, horizontal asymptotes can be crossed
Example Graph: f(x) = (2x² + 1)/(x² + 1)
Theorems on Limits
1. Constant Rule: lim k = k
2. Constant Multiple Rule: lim k⋅f(x) = k⋅lim f(x)
3. Sum/Difference Rule: lim(f ± g) = lim f ± lim g
4. Product Rule: lim(f ⋅ g) = lim f ⋅ lim g
5. Quotient Rule: lim(f/g) = lim f / lim g, if denominator ≠ 0
6. Composition Rule: lim f(g(x)) = f(lim g(x)) if f is continuous at that point
7. Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) and lim f = lim h = L → lim g = L
Types of Discontinuities
• Removable discontinuity: a hole in the graph – Limit exists, but f(c) is undefined or not equal to limit
– Example: (x² – 1)/(x – 1)
• Jump discontinuity: left- and right-hand limits exist but are unequal – Step functions, piecewise functions
• Infinite discontinuity: vertical asymptotes – Limit approaches ±∞
Algebraic Limits
• Simplify algebraically when limit gives 0/0 form:
– Factor and cancel common terms
– Rationalize numerator if square roots are present
Continuity
• A function f(x) is continuous at x = c if:
1. f(c) is defined
2. limx→c f(x) exists
3. limx→c f(x) = f(c)
• Continuous on an interval → no jumps, breaks, or holes
• Polynomial, exponential, log, and trig functions are continuous on their domains
Intermediate Value Theorem (IVT)
• If f(x) is continuous on [a, b] and N is between f(a) and f(b), then ∃ c ∈ (a, b) such that f(c) = N
• Used to show a root exists (e.g., f(a) < 0, f(b) > 0 → sign change → root in (a, b))
Extreme Value Theorem (EVT)
• If f(x) is continuous on a closed interval [a, b], then f(x) has both a maximum and minimum value on [a, b]
In a Nutshell
Limits describe the behavior of a function near a point. A function is continuous if its value and limit match. Discontinuities (removable, jump, infinite) break this condition. Theorems like IVT and EVT rely on continuity to guarantee values exist in a given interval. Mastering limits builds the foundation for derivatives and integrals.