IndietroLimits and Continuity: Business Calculus Study Notes
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Limits and Continuity
Computing Limits Algebraically
Limits are fundamental to calculus, describing the behavior of functions as inputs approach specific values. Several properties allow us to compute limits algebraically.
Constant Rule: The limit of a constant is the constant itself.
Power Rule: The limit of a function raised to a power equals the power of the limit.
Root Rule: The limit of a root equals the root of the limit.
Sum/Difference Rule: The limit of a sum or difference is the sum or difference of the limits.
Product Rule: The limit of a product is the product of the limits.
Quotient Rule: The limit of a quotient is the quotient of the limits, provided the denominator's limit is not zero. , if
Constant Multiple Rule: The limit of a constant times a function is the constant times the limit.
Example:
Compute : Substitute : .
Compute : Substitute : .
Compute : Substitute : .
Limits Involving Indeterminate Forms
Sometimes, direct substitution yields an indeterminate form such as . In these cases, algebraic manipulation (like factoring) is necessary.
Factoring: Factor numerator and denominator to simplify the expression and remove the indeterminate form.
Example:
Compute : Factor numerator: Factor denominator: Cancel : Substitute :
Limits at Infinity
Limits as approaches infinity describe the end behavior of functions. For rational functions, the degree of the numerator and denominator determines the limit.
Basic Rule: and
General Rule: If is a constant and is a positive integer:
Example:
Compute : Divide numerator and denominator by : As , , so limit is $1$.
Compute : Highest degree in denominator is , so limit is $0$.
Definition of Continuity
Continuity describes whether a function has any breaks, jumps, or holes at a point or over an interval.
Continuous at a Point: A function is continuous at if:
is defined
exists
Discontinuity: If any of the above conditions fail, has a discontinuity at .
Continuous on an Interval: is continuous on an open interval if it is continuous at every point in .
Everywhere Continuous: is continuous on .
Example:
Function is not defined at (division by zero), but for , . The limit as is $4f(2)fx = 2$.
Types of Discontinuity
Discontinuities can be classified as follows:
Type | Description | Example |
|---|---|---|
Removable | Hole in the graph; limit exists, but function is not defined or not equal to limit at that point. | at |
Jump | Function jumps from one value to another; left and right limits exist but are not equal. | Piecewise function with different values at a point |
Infinite | Function approaches infinity at a point; limit does not exist. | at |
Continuity of Piecewise Functions
For piecewise functions, check continuity at the points where the formula changes.
Evaluate left and right limits at the transition point.
Check if the function value matches the limit.
Example:
Let Check continuity at : Left limit: Right limit: Since left and right limits are not equal, is not continuous at .
Practice Problems
Compute limits using algebraic rules and factoring.
Discuss continuity for various functions, including rational, root, and piecewise functions.
Example:
Is continuous at ? For , . At , . Limit as is . Since , is not continuous at .
Additional info: These notes expand on the lecture's brief points, providing definitions, examples, and formulas for limits and continuity, including handling indeterminate forms and piecewise functions.
