Overview
This lecture introduces the concept of limits in calculus, explains how to evaluate them analytically and graphically, and covers different types of discontinuities.
Introduction to Limits
- A limit describes the value a function approaches as the input approaches a certain point.
- Direct substitution involves plugging the value directly into the function to check the output.
- If substitution gives 0/0 (undefined), alternative methods are required.
Analytical Techniques for Evaluating Limits
- Substitute values close to the target to estimate the limit if direct substitution fails.
- Factoring can simplify functions and eliminate zero denominators.
- For difference of cubes: ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) ).
- Complex fractions can be simplified by multiplying numerator and denominator by the least common denominator (LCD).
- For expressions with radicals, multiply numerator and denominator by the conjugate to simplify.
Examples of Analytical Evaluation
- Factoring and canceling ( x-2 ) from ( (x^2-4)/(x-2) ) gives limit as 4 as ( x \to 2 ).
- Substituting directly into ( x^2 + 2x - 4 ) at ( x = 5 ) gives limit 31.
- Applying difference of cubes and factoring ( (x^3-27)/(x-3) ) gives limit 27 as ( x \to 3 ).
- Simplifying complex fractions and using substitution yields results, e.g., ( \lim_{x \to 3} \frac{1/x - 1/3}{x-3} = -1/9 ).
- Using the conjugate for ( (\sqrt{x}-3)/(x-9) ) as ( x \to 9 ), the limit is 1/6.
- Combining LCD and conjugate for ( \frac{1/\sqrt{x} - 1/2}{x-4} ) as ( x \to 4 ) gives limit ( -1/16 ).
Evaluating Limits Graphically
- To find limits graphically, identify the y-value as the input approaches from left or right.
- One-sided limits: approach from only one direction (left or right).
- If left and right limits differ, the two-sided limit does not exist.
- Function value is given by the y-value at the closed (filled-in) circle on the graph.
Types of Discontinuities
- Jump discontinuity: the graph jumps to a different value (not removable).
- Hole (removable discontinuity): single missing point due to simplification.
- Infinite discontinuity: asymptote, function diverges (not removable).
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a point.
- Direct Substitution — Replacing the variable with a specific value to evaluate a limit.
- Complex Fraction — A fraction with fractions in the numerator, denominator, or both.
- Conjugate — For an expression ( a - b ), the conjugate is ( a + b ).
- Jump Discontinuity — A sudden change in the value of the function.
- Removable Discontinuity — A hole that can be "filled" by redefining the function at that point.
- Infinite Discontinuity — Occurs at vertical asymptotes where the function diverges.
Action Items / Next Steps
- Practice evaluating limits using direct substitution, factoring, common denominators, and conjugates.
- Try out graphical examples to better understand one-sided and two-sided limits.
- Review the difference between removable and non-removable discontinuities.