Calculus: Limits Graphically

A limit is the __________ a function ______________ from both the left and the right side of a given __________.

A one‐sided limit is the __________ a function approaches as you approach a given __________ from either the ______ or ______ side.

What is the common notation used for a limit?

What does 'lim f(x) as x approaches 3' signify?

What is a limit in terms of approaching a point?

From which two sides can a one-sided limit be approached?

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.

This quiz focuses on understanding limits in calculus, specifically exploring what limits are and how they can be analyzed graphically. You will encounter questions that require you to think about the behavior of functions approaching certain values from both sides. Test your knowledge and hone your skills in this essential calculus concept.