Introduction to Limits

• Definition of Limits: A fundamental concept in calculus where we determine what a function approaches as the input (x) approaches a certain value. The limit of a function describes the behavior of that function near a specific point, even if the function itself is not defined at that point.

• Importance: Limits are critical for understanding continuity, derivatives, and integrals in calculus. They allow us to analyze the behavior of functions, make predictions about their values, and solve problems involving instantaneous rates of change.

Example 1: Function Definition and Discontinuity

• Function Definition: Let f(x) = (x - 1) / (x - 1) simplifies to f(x) = 1, except at x = 1. While the function appears consistent everywhere except one point, this emphasizes the role of limits in defining function behavior over specific inputs that are undefined.

• Undefined Point: At x = 1: f(1) results in 0/0 which is undefined. The concept of indeterminate forms arises here, illustrating a key aspect of limits. The function is not defined at this point due to division by zero, making the evaluation at this point impossible.

Graphing f(x)

• Graph Behavior: For x ≠ 1, f(x) approaches 1. The gap created at x = 1 is crucial, as it signifies a point of discontinuity.

• Visual Representation: The graph appears as a horizontal line at y = 1, with an open circle at (1, 1) indicating the undefined nature of the function at this point. This visual representation highlights the limit's role in understanding function behavior across various input values.

Limit of f(x) as x approaches 1

• Limit Statement:

  [ \lim_{x \to 1} f(x) = 1 ]

• Explanation: As x gets closer to 1 from either direction, f(x) remains close to 1. This reinforces how limits provide insights into function behavior, even when direct substitution fails due to undefined points.

Example 2: Another Function with Discontinuity

• New Function Definition: Let g(x) = x² when x ≠ 2; g(2) = 1. This creates a scenario where limits can be leveraged to evaluate function behavior that diverges from traditional definitions.

• Graph Behavior: The function appears as a parabolic curve of y = x² everywhere except at x = 2, where it drops to y = 1. The existence of a jump at this specific input illustrates a different form of discontinuity - a removable discontinuity.

Graphing g(x)

• Continuity Analysis: Displays a parabola with a noticeable gap at x = 2. At x = 2, the function value shifts from 4 (value of the parabola) to a defined point of 1. This discontinuity can hinder our interpretation of the function without limits.

Limit of g(x) as x approaches 2

• Limit Statement:

  [ \lim_{x \to 2} g(x) = 4 ]

• Function Evaluation: While g(2) = 1, this isolated value does not affect the overall limit calculated based on surrounding values. The limit focused on nearby points in the domain provides a clearer understanding of function behavior than direct evaluation at the discontinuity.

• Approaching from Both Directions: Values near 2, regardless of the side they approach from, approach closer to 4:

  • From the left: evaluating at 1.9 (3.61), 1.99 (3.96), all converging towards 4.

  • From the right: evaluating at 2.1 (4.41), 2.01 (4.04), also converging towards 4. This showcases the core principle of limits being contingent upon nearby values rather than the function's defined value at that point.

Calculation Verification

• Numeric Approach: Using a calculator to verify limits as x approaches 2: evaluating values such as 1.999999 (resulting in 3.999998) reinforce the notion of convergence towards 4 as x nears 2, leading to precise insights regarding limits in the context of x².

Summary and Conclusion

• Key Takeaways: Limits play a vital role in understanding the behavior of functions at points of discontinuity. They provide clarity concerning function values and assist in evaluating pieces of functions that may not behave typically at specific points.

  • Furthermore, the limits can often exist even when the function itself is undefined at that point, offering a powerful tool for calculus problems.

• Real-World Application: Understanding limits is essential for further study in calculus concepts such as derivatives and integration, as they form the foundation for advanced mathematical concepts and applications in physics, engineering, and other fields that rely on continuous change.