Overview
This lecture introduces the concept of limits in calculus, demonstrating analytic and graphical techniques to evaluate different types of limits and discussing discontinuities.
Understanding Limits Analytically
- A limit calculates the value a function approaches as the input approaches a specific point.
- Direct substitution is used if plugging in the value does not lead to an undefined expression.
- If substitution gives 0/0, try plugging in values very close to the target to estimate the limit.
- Factoring expressions can help remove zero denominators and allow direct substitution.
- For complex fractions, multiply numerator and denominator by the common denominator to simplify.
- For differences of cubes, use the formula (a^3 - b^3 = (a-b)(a^2 + ab + b^2)).
- For functions with square roots, multiply by the conjugate to rationalize the expression.
Evaluating Complex and Radical Limits
- Simplify complex fractions by multiplying by the least common denominator.
- Use conjugates to handle radicals in numerator or denominator by multiplying top and bottom by the conjugate.
- Always check for factors that can be canceled before substituting the limiting value.
Limits Involving Substitution Examples
- Example: (\lim_{x \to 2} \frac{x^2-4}{x-2} = 4) after factoring and canceling (x-2).
- Example: (\lim_{x \to 5} (x^2 + 2x - 4) = 31) by direct substitution.
- Example: (\lim_{x \to 3} \frac{x^3-27}{x-3} = 27) using difference of cubes.
Limits with Square Roots and Conjugates
- Example: (\lim_{x \to 9} \frac{\sqrt{x}-3}{x-9} = \frac{1}{6}) after multiplying by conjugate.
- Example: (\lim_{x \to 4} \frac{1/\sqrt{x} - 1/2}{x-4} = -\frac{1}{16}) after complex fraction simplification and conjugate use.
Evaluating Limits Graphically
- To find a limit graphically, approach the vertical line (x-value) from the left and right.
- If both sides approach the same y-value, the limit exists; if not, it does not exist.
- The function value at a point is found at the closed circle on the graph.
- Vertical asymptotes (e.g., (x=3) for (1/(x-3))) indicate undefined function values and non-existing limits as y approaches infinity or negative infinity.
Types of Discontinuities
- Jump discontinuity: graph jumps at a point; not removable.
- Removable (hole) discontinuity: a single point missing from the curve, can be removed by redefining the function.
- Infinite discontinuity: function approaches infinity near a vertical asymptote; not removable.
Key Terms & Definitions
- Limit — the value a function approaches as the input approaches a specific number.
- Direct substitution — plugging the target value directly into the function to find the limit.
- Removable discontinuity — a hole in the graph where the function is undefined but can be redefined.
- Jump discontinuity — a sudden jump in the graph, creating a gap that cannot be removed.
- Infinite discontinuity — point where the function increases or decreases without bound (vertical asymptote).
- Conjugate — the matching expression used to eliminate radicals, e.g., for (\sqrt{x}-a), the conjugate is (\sqrt{x}+a).
- One-sided limit — limit as (x) approaches from only the left or right side.
Action Items / Next Steps
- Practice evaluating limits using factoring, conjugates, and simplification.
- Review graphical interpretation of limits and discontinuities.
- Complete example problems provided in the lecture, especially those involving complex fractions and radicals.