Limits-of-a-Function

Introduction to Limits

  • Overview: Understanding limits is fundamental in calculus.

Example 1 - Finding a rectangle of Maximum Area

  • Problem Statement: Given 24 inches of wire, determine the dimensions for a rectangle with the largest area.

    • Variables: Let W represent the width and / represent the length of the rectangle.

    • Relationship: 2W + 2/ = 24 leads to / = 12 - W.

    • Area Function: A = /W = (12 - W)W = 12W - W².

Example 1 - Solution

  • Experimentation: By substituting various values of W into A = 12W - W²:

    • Observed that maximum area occurs at W = 6 inches.

    • Area Table:

      • Width W and corresponding Area A:

        • W: 5.0, A: 35.0

        • W: 5.5, A: 35.75

        • W: 5.9, A: 35.99

        • W: 6.0, A: 36.0

        • W: 6.1, A: 35.99

        • W: 6.5, A: 35.75

        • W: 7.0, A: 35.00

  • Conclusion: Limit of A as W approaches 6 is 36, denoted as lim (A) = 36.

What is a Limit?

  • Definition: Limits explore values a function approaches as its variable approaches a certain point.

Limits - An Informal Approach

  • Significance: Limits are crucial for understanding calculus which deals with change.

    • Examples of applications:

      • Acceleration of a rocket

      • Volume of a ship that is expanding

Another Fundamental Concept of Limits

  • Limits help evaluate behavior in very small regions around a point.

    • Fundamental calculus definitions rely on limits.

Example 2 - Limit of a Function

  • Function considered: f(x) = (16 - x²) / (4 + x), undefined at x = -4.

  • Function behavior near -4:

    • f(-4.1) = 8.1, f(-4.01) = 8.01, f(-4.001) = 8.001

    • f(-3.9) = 7.9, f(-3.99) = 7.99, f(-3.999) = 7.999

  • Conclusion: as x approaches -4, f(x) approaches 8, thus limit is 8.

Limit Notation

  • Approaching notation:

    • x -> a indicates x approaches a from the left.

    • x -> a+ indicates x approaches a from the right.

    • x -> a denotes approaching a from both sides.

Example 3 - Estimating a Limit Numerically

  • Function: f(x) = 3x - 2.

  • Values around x = 2:

    • x: 1.9, 1.99, 1.999, 2.0, 2.001, 2.01, 2.1

    • Corresponding f(x): 3.7, 3.97, 3.997, ?, 4.003, 4.03, 4.3

  • Conclusion: Limit as x approaches 2 is estimated to be 4.

Example 4 - Estimating a Limit Graphically

  • Consider f(x) = √(x + 1) - 1 as x approaches 0.

  • Conclusion: Despite f(x) being undefined at x = 0, limit exists.

One-sided Limits

  • Left-hand limit: If f(x) approaches L₁ from the left as x approaches a, we write: lim f(x) = L₁ as x -> a.

  • Right-hand limit: If f(x) approaches L₂ from the right as x approaches a, written as lim f(x) = L₂ as x -> a+.

Two-sided Limits

  • Definition: If left-hand limit and right-hand limit exist and are equal (L), write: lim f(x) = L as x -> a.

Existence and Nonexistence of Limits

  • Limit existence depends on definitions around the value of a, not on the function's value at a.

  • Example illustrating this: f at x = -4 can be 5, while the limit can still be 8.

Conditions Under Which Limits Do Not Exist

  1. Different values approached from left and right.

  2. Function increases or decreases without bound.

  3. Oscillation between two fixed values.

Example 5 - Comparing Left and Right Behavior

  • Analyzing graphs to determine non-existence of limits based on differing values from left and right approach.

Final Notes

  • Use given graphs to find values or determine non-existence of limits through visual interpretation.