Overview
This lecture introduces the fundamental concept of limits in calculus, explains why algebra alone cannot describe all function behaviors, and distinguishes between function values and the behavior of functions near specific points.
Algebra vs. Calculus: The Limits Motivation
- Precalculus and algebra focus on function values and domains—where functions are defined.
- Calculus examines not just if ( f(2) ) exists, but also how ( f(x) ) behaves as ( x ) approaches 2.
- Multiple functions can be undefined at a point but behave differently near that point (hole, asymptote, or jump).
Limit Concept Introduction
- A limit describes the behavior of function values as ( x ) approaches a specific value.
- The value of ( f(2) ) alone cannot capture how the function behaves around ( x = 2 ).
- The concept of a neighborhood considers values of ( x ) on both sides of 2, not just at 2.
Left and Right Limits
- The left-hand limit: behavior as ( x ) approaches 2 from values less than 2, written ( \lim_{x \to 2^-} f(x) ).
- The right-hand limit: behavior as ( x ) approaches 2 from values greater than 2, written ( \lim_{x \to 2^+} f(x) ).
- If both left and right limits are equal, the (two-sided) limit exists at that point.
Graphical Examples and Problems
- A function may have a "hole" (removable discontinuity) where the function is undefined but the limit exists.
- A function may have a vertical asymptote where the limits from left and right go to ( -\infty ) or ( +\infty ).
- "Jumps" in the graph cause the left and right limits to be unequal, so the limit does not exist at that point.
- It is possible for ( f(2) ) to exist while the limit does not, or vice versa.
Key Terms & Definitions
- Limit — The value that ( f(x) ) approaches as ( x ) gets close to a point, not necessarily the value at that point.
- Left-Hand Limit — The value approached by ( f(x) ) as ( x ) comes from the left (( x \to a^- )).
- Right-Hand Limit — The value approached by ( f(x) ) as ( x ) comes from the right (( x \to a^+ )).
- Removable Discontinuity — A "hole" in the graph where the limit exists but the function is undefined.
- Vertical Asymptote — A line ( x = a ) where ( f(x) ) increases or decreases without bound as ( x ) approaches ( a ).
- Jump Discontinuity — Occurs when the left and right limits at a point are not equal.
Action Items / Next Steps
- Review the concept of limit and the difference between evaluating ( f(a) ) and ( \lim_{x \to a} f(x) ).
- Prepare to study formal definitions and more examples of limits in the next class.