Calculus I: Limits, Continuity, and Secant Slopes – Study Guide

Limits and Continuity

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for analyzing continuity, derivatives, and the behavior of functions near points of interest.

• Definition: The limit of a function f(x) as x approaches a is the value that f(x) gets closer to as x gets closer to a.

• Notation: $\lim_{x \to a} f(x)$

• One-sided limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

• Existence: The limit exists if both one-sided limits are equal.

• Example: $\lim_{x \to 2} (x^2 + 2x + 3)$

Evaluating Limits Algebraically

Limits can often be evaluated by direct substitution, factoring, or rationalizing. When direct substitution leads to an indeterminate form (such as 0/0), algebraic manipulation is required.

• Direct Substitution: If f(x) is continuous at a, then $\lim_{x \to a} f(x) = f(a)$.

• Factoring: Factor numerator and denominator to cancel common terms.

• Example: $\lim_{x \to 2} \frac{x^2 - 3x + 2}{x - 2}$ Factor numerator: $(x-1)(x-2)$, so limit simplifies to $x-1$ as $x \to 2$, giving 1.

• Special Limits: $\lim_{n \to \infty} \frac{4n^2 + 2n}{n}$ Simplifies to $4n + 2$ as $n \to \infty$, which diverges.

Limits from Graphs

Limits can be estimated from graphs by observing the behavior of the function as x approaches a particular value.

• Key Points: Look for jumps, holes, or asymptotes.

• Example: If the graph approaches a value from both sides as x approaches 2, that value is the limit.

Delta-Epsilon Definition of Limit

The formal definition of a limit uses the concepts of epsilon ($\epsilon$) and delta ($\delta$) to rigorously prove that a function approaches a certain value.

• Definition: $\lim_{x \to a} f(x) = L$ means that for every $\epsilon > 0$, there exists $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

• Example: To prove $\lim_{x \to 1} 3x + 2 = 5$, for $\epsilon = 0.3$, find $\delta$ such that $|3x + 2 - 5| < 0.3$ whenever $|x - 1| < \delta$.

• Application: For $\epsilon = 0.3$, $|3x - 3| < 0.3 \implies |x - 1| < 0.1$ so $\delta = 0.1$ suffices.

Secant Slopes and Difference Quotients

Secant Line Slope

The slope of the secant line between two points on a function represents the average rate of change between those points.

• Formula: For points $(x_1, f(x_1))$ and $(x_2, f(x_2))$, the slope is $m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Example: Find the slope between $(-2, f(-2))$ and $(1, f(1))$.

Difference Quotient

The difference quotient is used to approximate the derivative of a function at a point.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Application: For $f(x) = x^2 + 2x + 3$, $x = 2$, $h = 0.01$: $\frac{f(2+0.01) - f(2)}{0.01}$

Average Rate of Change

Definition and Application

The average rate of change of a function over an interval quantifies how much the function's output changes per unit input.

• Formula: $\frac{f(b) - f(a)}{b - a}$

• Example: Using the "living wage" jobs table, calculate the average rate of change from 1998 to 2001: $\frac{200 - 75}{2001 - 1998} = \frac{125}{3} \approx 41.67$ jobs per year.

Continuity

Definition of Continuity

A function is continuous at a point if its value, its limit, and its behavior from both sides all agree at that point.

• Three Conditions:

  1. $f(a)$ is defined

  2. $\lim_{x \to a} f(x)$ exists

  3. $\lim_{x \to a} f(x) = f(a)$

• Discontinuity: If any of the above conditions fail, the function is not continuous at that point.

• Example: For a piecewise function, check each condition at the point where the definition changes.

Piecewise Functions

Definition and Analysis

Piecewise functions are defined by different expressions over different intervals. Continuity and limits at the boundaries must be checked carefully.

• Example: $f(x) = \begin{cases} mx - 10 & x < -2 \\ \frac{1}{x^2 + 3x - 2} & x \geq -2 \end{cases}$

• Check continuity at $x = -2$ by evaluating left and right limits and function value.

Tables: Living Wage Jobs Data

Tabular Data Interpretation

Tables are often used to present data for rate of change calculations and function analysis.

Year	1997	1998	1999	2000	2001
Jobs	85	95	99	75	200

Main Purpose: To calculate average rates of change and analyze trends over time.

Delta-Epsilon Proofs

Formal Limit Proofs

Delta-epsilon proofs are used to rigorously establish the value of a limit.

• Process: Given $\epsilon > 0$, find $\delta > 0$ such that $|f(x) - L| < \epsilon$ whenever $|x - a| < \delta$.

• Example: For $\lim_{x \to 1} 3x + 2 = 5$, if $\epsilon = 0.3$, then $|3x + 2 - 5| = |3x - 3| = 3|x - 1| < 0.3 \implies |x - 1| < 0.1$ so $\delta = 0.1$.

• Generalization: For $\lim_{x \to 3} (x + 8) = 11$, $|x + 8 - 11| = |x - 3| < \epsilon$, so $\delta = \epsilon$.

Summary Table: Continuity Conditions

Condition	Description
$f(a)$ is defined	The function has a value at $x = a$
$\lim_{x \to a} f(x)$ exists	The left and right limits at $x = a$ are equal
$\lim_{x \to a} f(x) = f(a)$	The limit equals the function value at $x = a$

Additional info:

• Some questions involve graphical analysis and piecewise functions, which are common in Calculus I.

• Delta-epsilon proofs and difference quotients are foundational for understanding derivatives and continuity.