Chapter 13: Limits: A Preview of Calculus
Calculus begins with the limit — the fundamental concept capturing how a function behaves as its input approaches a value. This chapter introduces limits numerically, graphically, and algebraically, and connects them to the derivative. This is the bridge from precalculus to calculus.
Textbook alignment
Sections
13.1Finding Limits Numerically and Graphically
Before any algebra, we understand limits intuitively: what value does f(x) approach as x gets close to a? Tables of values and graphs give numerical and visual insight into limit behavior.
13.2Finding Limits Algebraically
Limit laws allow algebraic computation. For continuous functions, direct substitution works. For 0/0 indeterminate forms, we factor, rationalize, or simplify before substituting. For limits at infinity, we divide by the highest power of x.
13.3The Derivative as a Limit
The derivative is the limit of average rates of change as the interval shrinks to zero. Geometrically it is the slope of the tangent line. This section is the capstone of precalculus and the entry point to all of calculus.
13.4Limits at Infinity; Limits of Sequences
Limits at infinity describe end behavior — what happens as x → ±∞. These connect directly to horizontal asymptotes. For sequences, the limit tells us if the sequence converges to a finite value or diverges.
What's included — free
- ✓Visual concept explanations with diagrams for every section
- ✓Step-by-step worked examples you can study at your pace
- ✓Key vocabulary and memory aids for each topic
- ✓Printable worksheets generated for each section
Upgrade for unlimited practice, private tutoring, study planner, and exam mode. View plans