BackBusiness Calculus: Limits and Continuity (Section 1.1 Study Notes)
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Limits and Continuity
Introduction to Limits
In calculus, the limit of a function describes the behavior of the function as the variable approaches, but does not equal, a specified number. Limits are foundational for understanding derivatives and continuity.
Definition: The limit of f(x) as x approaches c is L if the values of f(x) get arbitrarily close to L as x gets close to c (but not equal to c).
Notation:
Key Point: The limit describes the behavior near a point, not necessarily the value at the point.
Difference Between Value and Limit
It is important to distinguish between the value of a function at a point and the limit of the function as it approaches that point.
Value at a Point: is the actual value of the function at x = c.
Limit at a Point: is the value that f(x) approaches as x gets close to c, but not equal to c.
Example: If , is undefined, but can exist.
Evaluating Limits Graphically and Numerically
Limits can be evaluated using graphs, tables, or algebraic manipulation.
Graphical Approach: Observe the behavior of f(x) as x approaches c from both sides.
Numerical Approach: Use tables to see how f(x) behaves as x gets closer to c from the left and right.
Algebraic Approach: Simplify the function, if possible, to evaluate the limit directly.
One-Sided Limits
Sometimes, the behavior of a function as x approaches c from the left is different from the right. These are called one-sided limits.
Left-Hand Limit:
Right-Hand Limit:
Example: If approaches different values from the left and right at x = c, the two-sided limit does not exist.
Definition of Left and Right Limits
The left and right limits are defined as follows:
Left Limit: is the value f(x) approaches as x gets close to c from the left.
Right Limit: is the value f(x) approaches as x gets close to c from the right.
If both limits are equal, the two-sided limit exists: .
If not, does not exist (D.N.E.).
Continuity
A function is continuous at a point if there are no breaks, jumps, or holes in its graph at that point. Formally:
Definition: f is continuous at x = a if and only if .
Three conditions for continuity at x = a:
exists
is defined
Types of Discontinuity:
Jump Discontinuity: The function jumps from one value to another.
Hole Discontinuity: The function is not defined at a point, but the limit exists.
Summary Table: Continuity at a Point
a | f(a) | lim f(x) | Continuous Yes/No |
|---|---|---|---|
1 | 2 | 2 | Yes |
2 | 3 | does not exist | No |
3 | 2 | 2 | Yes |
4 | undefined | 2 | No |
Limit Laws
The following rules govern the mathematics of limits:
(if the limit exists)
(if denominator is not zero)
Example Problems
Evaluating Limits from Graphs: Use the graph to determine by observing the behavior as x approaches c.
Using Tables: Create a table of values for x approaching c from both sides to estimate the limit.
Algebraic Manipulation: Simplify the function to remove indeterminate forms and evaluate the limit.
Piecewise Functions: For functions defined by different formulas on different intervals, check limits from both sides and continuity at the boundary points.
Exercises
Find limits using graphs, tables, and algebraic methods.
Identify points of discontinuity in given functions.
Determine continuity of piecewise-defined functions.
Additional info:
These notes cover Section 1.1 of a Business Calculus textbook, focusing on limits and continuity, which are essential for understanding calculus in business applications.
