Hey guys! Getting ready for your IAP Calculus AB Unit 1 exam? No sweat! This guide will walk you through everything you need to know to nail that test. We'll cover the key concepts, provide practice problems, and offer some tips and tricks to help you succeed. Let's dive in!

Understanding Limits: The Foundation of Calculus

Limits are the absolute bedrock upon which all of calculus is built. So, understanding limits is super important. In essence, a limit explores what happens to a function's output as its input gets really, really close to a specific value. It's not necessarily about what the function equals at that value, but rather the trend it follows. Think of it like approaching a destination. You might get incredibly close, but the limit is the destination itself, not necessarily the act of arriving. There are several ways to evaluate limits such as numerical, graphical, and algebraic methods. Numerical methods use tables of values to estimate the limit. Graphical methods involve analyzing the graph of the function near the point of interest. Algebraic methods involve manipulating the expression to simplify it and evaluate the limit directly. Understanding these methods is crucial for solving a variety of limit problems. Limits also form the basis for understanding continuity, which we'll discuss later. For example, if a function approaches a certain value as x approaches a, then the limit of f(x) as x approaches a exists. However, the limit may not always exist. For example, if the function approaches different values from the left and right sides of a, then the limit does not exist. It's also important to understand that the existence of a limit does not necessarily imply that the function is defined at that point. The function could have a hole or a jump discontinuity at that point. Knowing the different scenarios and how to identify them is key to mastering limits. Practice with various examples, and soon, limits will become second nature!

Methods for Evaluating Limits

• Numerical Method: Constructing tables of values to see the trend as x approaches a certain value. This is great for getting an initial idea of what the limit might be.
• Graphical Method: Analyzing the graph of the function. Look for holes, jumps, or asymptotes that might affect the limit.
• Algebraic Method: This is where you use techniques like factoring, rationalizing, and simplifying to evaluate the limit directly. This is often the most precise method.

Key Limit Laws to Remember

• Limit of a Constant: The limit of a constant is just the constant itself. lim (x→a) c = c
• Limit of a Sum/Difference: The limit of a sum or difference is the sum or difference of the limits. lim (x→a) [f(x) ± g(x)] = lim (x→a) f(x) ± lim (x→a) g(x)
• Limit of a Product: The limit of a product is the product of the limits. lim (x→a) [f(x) * g(x)] = lim (x→a) f(x) * lim (x→a) g(x)
• Limit of a Quotient: The limit of a quotient is the quotient of the limits (as long as the limit of the denominator isn't zero). lim (x→a) [f(x) / g(x)] = lim (x→a) f(x) / lim (x→a) g(x), provided lim (x→a) g(x) ≠ 0
• Limit of a Power: The limit of a function raised to a power is the limit of the function, raised to that power. lim (x→a) [f(x)]^n = [lim (x→a) f(x)]^n

Continuity: When Limits and Function Values Align

Continuity is all about whether you can draw a function without lifting your pencil. More formally, a function f(x) is continuous at a point x = a if three conditions are met:

1. f(a) is defined (the function has a value at that point).
2. lim (x→a) f(x) exists (the limit exists as x approaches a).
3. lim (x→a) f(x) = f(a) (the limit equals the function value at that point).

If any of these conditions fail, the function is discontinuous at x = a. It's essential to check all three conditions to determine continuity. There are three main types of discontinuities: removable, jump, and infinite. A removable discontinuity occurs when the limit exists but does not equal the function value, usually due to a hole in the graph. A jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal, resulting in a sudden jump in the graph. An infinite discontinuity occurs when the function approaches infinity as x approaches a certain value, leading to a vertical asymptote. Understanding these types of discontinuities helps in analyzing the behavior of functions and their limits. When you are asked about continuity on an interval, remember that a function is continuous on an open interval (a, b) if it is continuous at every point in the interval. For a closed interval [a, b], the function must also be continuous from the right at a and continuous from the left at b. This means the limit from the right at a must equal f(a), and the limit from the left at b must equal f(b). Knowing these details ensures you're covering all bases when assessing continuity. Functions can be continuous everywhere, or they can have points of discontinuity. Polynomials, for example, are continuous everywhere. Rational functions are continuous everywhere except where the denominator is zero. Piecewise functions can be tricky; you need to check continuity at the points where the pieces connect. Continuity is a critical concept in calculus because many theorems and applications rely on the assumption that functions are continuous. For example, the Intermediate Value Theorem and the Extreme Value Theorem both require continuous functions.

Types of Discontinuities

• Removable Discontinuity: A hole in the graph. You can