Overview
This lecture introduces the concept of limits in calculus, explains their importance in defining derivatives and integrals, and presents both intuitive and formal definitions, including key examples and their connection to physics.
The Importance of Limits in Calculus
- Limits are foundational in calculus, used to define the derivative and the definite integral.
- Understanding limits is essential for analyzing continuity and the behavior of functions near specific points.
Intuitive Understanding of Limits
- Approaching a value without ever reaching it illustrates the core idea behind limits (e.g., halving the distance to a wall indefinitely).
- Summing infinite series (such as repeatedly dividing a square) can result in a finite value, showing the power of limits.
Mathematical Definition and Notation
- The limit of a function describes its behavior as the input approaches a particular value, not necessarily at that value.
- Left and right-sided (lateral) limits describe approaching from below and above, respectively.
- If both lateral limits are equal, the two-sided limit exists and is unique.
Existence and Non-Existence of Limits
- A limit exists if both left and right lateral limits exist and are equal.
- Limits may not exist if left and right limits differ or tend toward infinity.
- When limits approach infinity, it expresses unbounded growth rather than the existence of a finite limit.
- Vertical asymptotes correspond to points where the function grows without bound as the input approaches a specific value.
Distance and Absolute Value in Limits
- The absolute value is used to represent the distance between two points on the number line and is always non-negative.
- The distance between points a and x is represented as |x - a|.
Formal (Epsilon-Delta) Definition of Limit
- For any chosen error bound (epsilon), however small, there exists a distance (delta) so that if 0 < |x - a| < delta, then |f(x) - L| < epsilon.
- This formalizes the concept that f(x) can be made arbitrarily close to L by taking x sufficiently close to a, without necessarily being equal to a.
Limits in Physics: The Speed of Light
- Special relativity uses limits, as certain quantities tend to infinity as speed approaches the speed of light.
- The Lorentz factor (Gamma) increases without bound as velocity approaches the speed of light, imposing a universal speed limit.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a specific point.
- Lateral Limit — The value a function approaches from one side (left or right) as the input nears a point.
- Epsilon (ε) — An arbitrarily small positive number representing allowable error in the function's value.
- Delta (δ) — A corresponding small distance from the input value ensuring the error is within epsilon.
- Asymptote — A line that the graph of a function approaches but never touches.
- Absolute Value — The non-negative distance between two numbers, denoted |x - a|.
- Lorentz Factor (Gamma) — A factor in relativity defined as ( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} ).
Action Items / Next Steps
- Review the epsilon-delta definition and practice applying it to simple functions.
- Solve examples calculating left, right, and two-sided limits, including cases where limits do not exist.
- Read about the application of limits in physics, especially in special relativity.