Jan-30_Cal

Section 3.2 Overview

  • Focus of the discussion: Limits and introduction to continuity.

Definition of Limits

  • Definition discussed in detail:

    • A limit represents the behavior of a function as the input approaches a specific value.

    • Formal expression: As x approaches a value c , f(x) approaches L . This is often written as:
      ext{lim}_{x o c} f(x) = L .

  • Key point: The limit concerns the values that f(x) approaches as x nears c , not necessarily the value of f(c) itself.

Previous Discussion Highlights

  • Graphical representation was emphasized as a tool in understanding limits.

  • Types of limits covered include:

    • Left-hand limits

    • Right-hand limits

    • Limits at infinity

    • Infinite limits

Evaluating Limits Without a Graph

  • Limit evaluation example: ext{lim}_{x o 1} (x^2 - x + 3) .

  • Initial suggestion: Plug in values close to 1 (e.g., 0, 2); noted these values were not appropriate for limit purposes.

  • Reminded that limits require values to approach a point closely, not directly evaluate the point itself.

  • Suggested a systematic approach for evaluating limits:

    1. Left-hand limit: Evaluate values slightly less than 1 (e.g., 0.9, 0.99).

    2. Right-hand limit: Evaluate values slightly more than 1 (e.g., 1.1, 1.01).

Left-hand Limit Calculations

  • Evaluated f(0.9) , f(0.99) :

    • For f(0.9) : (0.9^2 - 0.9 + 3) = 2.91 .

    • For f(0.99) : (0.99^2 - 0.99 + 3) = 2.9901 .

    • Observed trend toward 3 as inputs get closer to 1 from the left.

Right-hand Limit Calculations

  • Evaluated f(1.1) , f(1.01) :

    • For f(1.1) : (1.1^2 - 1.1 + 3) = 3.01 .

    • For f(1.01) : (1.01^2 - 1.01 + 3) = 3.001001 .

    • Observed trend toward 3 as inputs get closer to 1 from the right.

  • Conclusion after evaluating limits:

    • From both sides, the limits approach 3; thus, the overall limit exists and is equal to 3:
      ext{lim}_{x o 1} (x^2 - x + 3) = 3 .

Definition of Continuity

  • Discussion on what it means for a function to be continuous:

    • A function f is continuous at point c if:

    1. The function is defined at c : f(c) exists.

    2. The limit exists at c : ext{lim}_{x o c} f(x) exists.

    3. The value of the function equals the limit: ext{lim}_{x o c} f(x) = f(c) .

  • Misconception about continuity: Simply continuing visually (like a line) does not suffice mathematically.

  • The continuity at point c depends on the behavior of both one-sided limits and their equality:

    • Both left-hand limit and right-hand limit must converge to the same value.

Conditions for Continuity

  • Three criteria for continuity at a point c :

    1. Existence of the function at c :

    • f(c) must be defined.

    1. Existence of the limit as x approaches c :

    • ext{lim}_{x o c} f(x) must exist (the left-hand limit equals the right-hand limit).

    1. Equality of the limit and function value:

    • ext{lim}_{x o c} f(x) = f(c) .

Analyzing Specific Values for Continuity

  • Given the expression, students tasked with determining where the function is continuous or not continuous:

    • Example values provided and analyzed included:

    • x = -5 : Continuity question reveal failed limit condition.

    • x = -3 : Limit exists but is not equal to f(c) (fails third criteria).

    • x = -1 : Identified failure in limit existence (different left/right limits).

    • x = 1 : Existential failure for function value (no output at c ).

    • For each case, reinforced connection back to the three criteria for continuity and iterating that if one criteria fails, the function fails continuity.

Functions that are Continuous

  • Acknowledge types of continuous functions:

  1. Polynomials:

    • Defined to be a function made up of terms of the form a_n x^n , where n is a non-negative integer.

  2. Rational Functions:

    • Defined as the ratio of two polynomial functions.

    • Continuous as long as the denominator does not equal zero.

    • Must be analyzed carefully especially where denominators may be zero to ensure limits can be evaluated and continuity verified.

Example of Rational Function

  • Evaluated limits using rational functions to illustrate continuity and where it fails due to division by zero. Emphasis on need for careful evaluation of continuity related to the function behavior (factoring, etc.).

Wrap-Up and Quiz Introduction

  • Preparation for a quiz related to limits implemented post-discussion.

  • Emphasis on practical application of the discussed concepts.

  • Structure of the quiz included graphical components with associated limit questions.