AP Calculus: Limits and Continuity

This chapter introduces the concept of a limit, which is the foundational idea of all of calculus. A limit describes the value a function approaches as its input gets arbitrarily close to a specific point. You will learn to find limits by examining graphs, analyzing tables of values, and using algebraic techniques. The main sub-topics are one-sided limits, limits involving infinity (which relate to horizontal and vertical asymptotes), and the formal properties of limits.

The chapter then uses limits to define continuity. A function is continuous if its graph can be drawn without lifting your pencil. You will learn the three-part definition of continuity at a point and how to identify and classify different types of discontinuities, such as removable (holes), jump, and infinite. Key theoretical results like the Squeeze Theorem and the Intermediate Value Theorem are also covered.

The most common difficulty is in evaluating limits that result in an indeterminate form like 0/0. These problems require strong algebraic manipulation skills, including factoring, multiplying by conjugates, and simplifying complex fractions. The chapter relies heavily on a solid understanding of functions and graphs from pre-calculus.

Formulas for AP Calculus: Limits and Continuity

Limit of a Quotient: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$ This property holds provided that the limit of the denominator is not zero.
Squeeze Theorem: $g(x) \le f(x) \le h(x) \text{ and } \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L \implies \lim_{x \to c} f(x) = L$ Used to find the limit of a function that is 'squeezed' between two other functions.
Special Trigonometric Limit (Sine): $\lim_{x \to 0} \frac{\sin x}{x} = 1$ This is essential for finding limits of many trigonometric functions.
Special Trigonometric Limit (Cosine): $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ Often used in combination with the sine limit to resolve indeterminate forms.
Definition of Continuity at a Point: $\lim_{x \to c} f(x) = f(c)$ This single statement implies three conditions: f(c) is defined, the limit exists, and they are equal.
Intermediate Value Theorem (IVT): $\text{If } f \text{ is continuous on } [a, b] \text{ and } N \text{ is between } f(a) \text{ and } f(b), \text{ then there exists a } c \text{ in } (a, b) \text{ such that } f(c) = N.$ Guarantees that a continuous function takes on every value between two points.
Limit Definition of a Derivative: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ Gives the slope of the tangent line to f(x) at the point x=a.
Alternate Limit Definition of a Derivative: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ An equivalent form used to find the slope of the tangent at x=a.