Overview
This lecture introduces the fundamental concept of limits in calculus, covering their definition and basic techniques for evaluating limits using substitution, factoring, and graphical analysis.
What is a Limit?
- A limit describes the value a function approaches as the input (x) approaches a specific value.
- Limits are foundational to understanding calculus concepts like continuity, derivatives, and integrals.
- Limits can exist from both sides (left and right) or one side (one-sided limits).
Evaluating Limits: Methods
- Direct Substitution: Plug the value into the function; if the result is defined, this is the limit.
- If direct substitution results in an indeterminate form (like 0/0), other methods must be used.
- Factoring: Factor numerator and denominator to cancel common terms, then substitute the value.
- Rationalizing: Multiply by a conjugate if radicals are present to simplify complex fractions.
- Graphical Approach: Observe the function’s graph to see where y-values approach as x nears the target value.
Special Limit Types & Examples
- Complex Fractions with Radicals: Use rationalizing technique to evaluate limits that involve square roots or other roots.
- Vertical Asymptotes: If function shoots to ±∞ as x approaches a value, the limit does not exist (DNE) there.
- One-Sided Limits: Evaluate the limit as x approaches from only one side (left or right) if needed.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches some point.
- Direct Substitution — Evaluating a limit by simply plugging in the value for x.
- Indeterminate Form — A mathematical expression such as 0/0 where the limit cannot be determined directly.
- Vertical Asymptote — A line x = a where a function grows without bound as x approaches a.
- One-Sided Limit — The value the function approaches as x comes from only one side (left or right).
Action Items / Next Steps
- Practice evaluating limits using substitution, factoring, and rationalizing techniques.
- Review graphical examples to understand limits visually.
- Complete any assigned problems on basic limit evaluation.