Continuity

Limits

Warm-up Examples: Review of Limits

Example 1: Calculating Limits

• Objective: Find the following limits.

• Function provided: 2x + 3 - 4x^2

• Limit expression: \lim_{x \to -\infty} \frac{3x + 1}{-4x^2}

• Result: \lim_{x \to -\infty} \frac{-4x}{3}

Example 2

• Given: \lim_{x \to -\infty} \frac{-4x}{3}

• Understanding the behavior of functions as x approaches -\infty.

Understanding Limits

Definitions and Concepts

• Limit: The limit of a function as x approaches a certain value is not the same as substituting that value into the function. Despite this, both processes often yield the same value.

• Continuity: A function is defined to be continuous at a point c if it meets the following conditions:

  • The function has a value at c: f(c) exists.

  • The limit of the function as it approaches c exists: \lim_{x \to c} f(x) exists.

  • These two values are equal: f(c) = \lim_{x \to c} f(x).

• Point of Discontinuity: Any point at which a function is not continuous.

Graphical Interpretation

• Graphing a function helps identify points of discontinuity visually. The graph should be analyzed to determine where the function is undefined or does not connect smoothly.

Continuity in Functions

General Observations on Continuity

• Many functions are continuous at every point where they are defined. Notably, a function cannot be continuous at points where it is undefined.

• Types of functions that are typically continuous where defined include:

  • Polynomials: Functions in the form an x^n + a{n-1} x^{n-1} + … + a0 where an are constants and n is a non-negative integer.

  • Rational Functions: Quotients of polynomials.

  • Radicals: Root functions, e.g., square root functions.

  • Exponential Functions: Functions of the form f(x) = a^x for a constant a > 0.

  • Logarithmic Functions: The inverse of exponential functions.

  • Absolute Value Functions: Functions that return the absolute value of the input.

  • Combinations: Combinations of the above functions.

• For the mentioned types of functions, identifying points of continuity corresponds directly to determining the function's domain.

Piecewise-Defined Functions

• Investigation of Continuity: It is crucial to examine piecewise-defined functions at transition points, where different formulas are applicable. To ensure continuity at these transition points, check:

  1. The limit from the left of the transition point.

  2. The limit from the right of the transition point.

  3. The value of the function at the transition point.

• Conditions for Continuity: If all three values (left limit, right limit, function value) are equal, the function is continuous at that point. If they are not equal, the function is discontinuous at that point.

Concept of One-Sided Continuity

• If the left limit exists and matches the function value, we say that the function is "continuous on the left" at that point.

• Conversely, if the right limit exists and matches the function value, the function is "continuous on the right".

• To establish overall continuity, the function must be continuous on both sides.

Examples of Discontinuity

• Example 1: Analyzing specific piecewise-defined functions for points of discontinuity.

• Example 2: Similar analysis for another piecewise function.

Continuity and Functions in Evaluating Inequalities

Importance of Continuity in Sign Changes

• Determining where a function f(x) is positive or negative is essential. This involves solving inequalities.

• Key Insight: A continuous function cannot change sign without passing through 0.

• To analyze the function, identify points where:

  • The function is discontinuous.

  • The function equals 0.

• The identified points create intervals on the number line where function behavior (sign) remains constant.

Constructing a Sign Chart

1. Find Partition Points: Identify all points where f(x) = 0 or the function is undefined, as these points partition the number line into intervals.

2. Testing Intervals: Select test numbers from each interval to ascertain the function's sign.

3. Conclusion: Document results for each interval, noting where the function is positive, negative, or zero.

• Example of Process: Analyzing the function given for points of discontinuity and constructing its sign chart systematically.