Overview
This lecture introduces the concept of limits, the foundation for calculus, and outlines the key differences between limits, function values, and the process for finding limits both graphically and algebraically, with emphasis on review of algebra skills.
Introduction to Calculus and Limits
- Calculus is split into differential calculus (derivatives: slopes/tangents) and integral calculus (areas under curves).
- Calculus was independently developed by Newton and Leibniz, each with distinct notations.
- Success in calculus relies heavily on strong algebra skills.
Understanding Limits: Concepts and Notation
- A limit describes what a function's output (y-value) approaches as the input (x-value) nears a specific value c.
- Notation: limₓ→c f(x) = L means as x approaches c, f(x) approaches L.
- The value of the limit does not necessarily equal the function’s value at that point.
Finding Limits Graphically
- To find a limit on a graph, observe the y-values as you approach the target x-value from both sides.
- The limit exists if both left and right sides approach the same y-value, regardless of the value at that exact point.
- If the left and right sides approach different values, the limit does not exist at that point (one-sided limits).
Finding Limits Algebraically
- Step 1: Plug in the target value for x.
- If you get a number, this is your limit.
- If you get 0/0 (an indeterminate form), factor/cancel or use conjugates, then plug in again.
- If you get any nonzero number over zero (e.g., 5/0), the limit does not exist (infinite discontinuity).
- For rational functions as x approaches infinity, divide all terms by the highest power of x in the denominator and apply limits.
Limits at Infinity and Asymptotes
- Limits as x approaches infinity often relate to the horizontal asymptotes of a graph.
- For rational functions, terms with powers of x in the denominator approach zero as x grows.
- If only x remains (e.g., x/1), the limit is infinity (limit does not exist).
Distinguishing Limits from Function Values
- The limit is about behavior near a point; the function value is the actual output at that point.
- They may differ, especially at points of discontinuity (breaks or holes in the graph).
Piecewise Functions and Break Points
- Graph each piece of a piecewise function separately and focus on break points (changeover x-values).
- Use open or closed circles to indicate strict or non-strict inequalities at breakpoints.
- Function value at a break point is where the closed dot lands; the limit considers values approaching from both sides.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a specified point.
- Derivative — The slope of the tangent line to a curve at a point (topic for later).
- Indeterminate Form — An expression like 0/0 where the limit can't be directly determined.
- Discontinuity — A break or hole in the graph where the limit and function value differ.
- Asymptote — A line that a graph approaches but never touches.
- Piecewise Function — A function defined by different expressions in different intervals.
Action Items / Next Steps
- Complete algebra review worksheets for foundational skills.
- Watch posted Math 130/College Algebra videos.
- Practice limit problems both graphically and algebraically.
- Review how to graph and analyze piecewise functions (see Algebra review resources).
- Optional: Review horizontal/vertical asymptotes and one-sided limits in pre-calc materials.