Calculus: Limits Graphically

Limits Overview

• A limit defines the value that a function approaches from both the left and right side of a specific point.
• Common notation for a limit is expressed as ( \lim_{x \to c} f(x) = L ), indicating the function ( f(x) ) approaches the value ( L ) as ( x ) approaches ( c ).

One-Sided Limits

• A one-sided limit examines the behavior of a function as it approaches a specific point from one side only.
• Left-hand limit: ( \lim_{x \to c^-} f(x) ) indicates the function approaches as ( x ) approaches ( c ) from the left.
• Right-hand limit: ( \lim_{x \to c^+} f(x) ) indicates the function approaches as ( x ) approaches ( c ) from the right.

Example Concepts

• Example of left-hand limit: Approaches the function value of 1 from the left side as ( x ) approaches 3.
• Example of right-hand limit: Approaches the function value of 2 from the right side as ( x ) approaches 3.

Formal Definition of Limits

• A limit ( L ) exists if ( f(x) ) can be made arbitrarily close to ( L ) by taking values of ( x ) sufficiently close to ( c ) (but not equal to ( c )).
• If the limit exists and is a real number, it confirms the function has a defined value at that point.

Visual Representation

• Graphs illustrate the behavior of a function as it approaches a certain point, highlighting the limits from both sides.