Calculus I: Limits, Continuity, and Derivatives – Unit 1 Study Guide

Unit 1: Limits, Continuity, and Derivatives

Limits – Graphically and Numerically

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach specific values. Understanding limits both graphically and numerically is essential for analyzing function behavior and preparing for more advanced topics.

• One-sided limits: The limit of a function as the input approaches a value from one side (left or right). Definition: The left-hand limit as $x \to a^-$ is $\lim_{x \to a^-} f(x)$; the right-hand limit as $x \to a^+$ is $\lim_{x \to a^+} f(x)$.

• Infinite limits: Occur when the function increases or decreases without bound as it approaches a certain value. Example: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$

• Numerical evaluation: Approximating limits by substituting values close to the point of interest. Example: To estimate $\lim_{x \to 2} f(x)$, calculate $f(1.9)$, $f(1.99)$, $f(2.01)$, $f(2.1)$ and observe the trend.

• Graphical evaluation: Observing the behavior of a function's graph near the point of interest to estimate the limit. Example: If the graph approaches a specific value as $x$ approaches $a$, that value is the limit.

• Existence of a limit: A limit exists at $x=a$ if both one-sided limits exist and are equal. Key Point: If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$, then $\lim_{x \to a} f(x) = L$.

Limits – Algebraically

Algebraic techniques allow for precise calculation of limits, including those involving infinity and cases where limits do not exist.

• Evaluating limits: Use substitution, factoring, rationalization, or special limit laws to find limits. Example: $\lim_{x \to 3} (x^2 - 9)/(x - 3)$ can be simplified to $\lim_{x \to 3} (x + 3)$, yielding $6$.

• Infinite limits: When the function grows without bound as $x$ approaches a value. Example: $\lim_{x \to 0} \frac{1}{x^2} = +\infty$

• Limits at infinity (algebraically): Describes the behavior of a function as $x$ approaches $+\infty$ or $-\infty$. Example: $\lim_{x \to \infty} \frac{2x^2 + 3}{x^2 - 1} = 2$

• Explaining non-existence of limits: Limits may not exist due to jump discontinuities, infinite oscillation, or unbounded behavior. Example: $\lim_{x \to 0} \sin(1/x)$ does not exist because the function oscillates infinitely as $x$ approaches $0$.

Continuity

Continuity describes functions that have no breaks, jumps, or holes at a given point or over an interval. Understanding continuity is crucial for analyzing function behavior and applying calculus techniques.

• Parameters for continuity: Finding values that make a piecewise function continuous at a point. Example: For $f(x) = \begin{cases} ax + 1 & x < 2 \\ 3x - 5 & x \geq 2 \end{cases}$, solve $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$ for $a$.

• Intervals of continuity: Determining where a function is continuous. Key Point: Polynomials are continuous everywhere; rational functions are continuous where the denominator is nonzero.

• Types of discontinuities: Classifying discontinuities as removable, jump, or infinite.

  • Removable: A single point is undefined or mismatched, but can be "fixed" by redefining the function.

  • Jump: The left and right limits exist but are not equal.

  • Infinite: The function approaches infinity at a point.

The Derivative

The derivative measures the rate of change of a function. It can be interpreted as the slope of the tangent line at a point, and is foundational for understanding motion, optimization, and change in calculus.

• Estimating average and instantaneous rates of change: Use tables or graphs to approximate the derivative. Average rate of change: $\frac{f(b) - f(a)}{b - a}$ Instantaneous rate of change: $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

• Graphical relationships: Given a graph of $f(x)$, sketch or interpret the graphs of $f'(x)$ (first derivative) and $f''(x)$ (second derivative). Key Point: Where $f(x)$ is increasing, $f'(x) > 0$; where $f(x)$ is concave up, $f''(x) > 0$.

• Application-based inference: Use context (such as motion, growth, or decay) to interpret the meaning of the derivative. Example: In physics, the derivative of position with respect to time is velocity.

Summary Table: Unit 1 Objectives and Parts

Objective	Parts
1: Limits - Graphically and Numerically Briggs 2.1 - 2.3	a. One-sided limits b. Infinite limits c. Numerically d. Graphically e. Determine whether a limit exists
2: Limits – Algebraically Briggs 2.3 - 2.5	a. Evaluate limits b. Infinite limits c. Limits at infinity, algebraically d. Explain why a limit doesn’t exist
3: Continuity Briggs 2.6	a. Determine values of parameters that make a function continuous b. Determine intervals of continuity of a function c. Identify the types of discontinuities
4: The Derivative Briggs 2.7, 3.1 - 3.2	a. Estimate average and instantaneous rates of change from a graph or table b. Given a graph, determine a graph for f, f', and/or f'' d. Make an inference from a function about the derivative, based on the application