Fall 2025 Chapter 2-1 and 2-2 Notes
Welcome to the notes for Chapter 2-1 and 2-2! In this section, we'll be exploring the fundamental concept of limits and how they relate to rates of change. We'll cover various techniques for evaluating limits, both numerically and graphically, and delve into the fascinating world of infinite limits.
Aleks Topics Covered:
- Estimating a limit numerically: We'll explore how to approximate the value of a limit by plugging in values that get progressively closer to a specific point. For example, consider estimating the limit of $\frac{(x+1)^3 - 1}{x}$ as $x$ approaches 0.
- Finding limits from a graph: Learn to read and interpret graphs to determine the limit of a function as $x$ approaches a particular value. Remember to check the limit from both the left and the right! If $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$, then $\lim_{x \to a} f(x) = L$.
- Infinite limits and graphs: Understand how to identify infinite limits from a function's graph, especially near vertical asymptotes. For example, if $f(x)$ approaches infinity as $x$ approaches $a$, we write $\lim_{x \to a} f(x) = \infty$.
- Infinite limits and rational functions: Learn to evaluate infinite limits for rational functions by analyzing the behavior of the numerator and denominator. Consider the function $f(x) = \frac{-x}{x-3}$. As $x$ approaches 3 from the left, the limit approaches $+\infty$, and as $x$ approaches 3 from the right, the limit approaches $-\infty$. Therefore, the limit as $x$ approaches 3 Does Not Exist (DNE).
- Finding the average rate of change of a function: Calculate the average rate of change using the formula $\frac{f(b) - f(a)}{b - a}$, which represents the slope of the secant line between two points on the function.
- Finding the slopes of several secant lines to a curve to estimate the slope of a tangent line: By calculating the slopes of secant lines with increasingly smaller intervals, we can estimate the slope of the tangent line at a specific point.
- Finding several average rates of change to estimate an instantaneous rate of change: The instantaneous rate of change is the limit of the average rate of change as the interval approaches zero. This can be represented as: $$ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$ This is also the derivative of the function $f(x)$.
Key Concepts and Examples
Let's consider an example to estimate a limit numerically. Suppose we want to find $\lim_{x \to 6} \frac{\sqrt{3x-14}-6}{x-6}$. By plugging in values close to 6, such as 5.999 and 6.001, we can observe that the limit appears to be around 2.25.
Another important concept is understanding limits from graphs. If the left-hand limit and the right-hand limit at a certain point are not equal, then the limit at that point does not exist. This often occurs at points of discontinuity or sharp turns in the graph.
Remember, practice is key! Work through plenty of examples to solidify your understanding of these concepts. Good luck, and don't hesitate to ask questions in class!