BackCalculus I: Limits, Continuity, and Tangents – Study Notes
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Limits and Their Evaluation
Definition and Basic Properties
Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for defining derivatives, continuity, and analyzing function behavior near points of interest.
Limit of a Function: The value that a function approaches as the input approaches a specific point.
Notation: denotes the limit of as approaches .
One-sided Limits: (from the left), (from the right).
Infinite Limits: Describes behavior as approaches infinity or negative infinity.
Example: Evaluate .
Direct substitution yields:
Special Limits Involving Trigonometric Functions
Trigonometric Limits: Often require the use of identities or the squeeze theorem.
Example:
Squeeze Theorem: Used when a function is bounded between two others whose limits are known.
Example: , with .
As , , so by the squeeze theorem, the limit is 0.
Limits Involving Indeterminate Forms
Indeterminate Forms: Expressions like or require algebraic manipulation or L'Hospital's Rule.
Example:
Example:
Example:
These limits may require factoring, rationalizing, or applying L'Hospital's Rule.
Continuity of Functions
Definition and Interval Notation
A function is continuous at a point if . Continuity on an interval means the function is continuous at every point in that interval.
Interval Notation: Used to describe where a function is continuous, e.g., .
Piecewise Functions: Functions defined by different expressions over different intervals.
Example: For , determine intervals of continuity.
Points of Discontinuity
Types: Removable, jump, and infinite discontinuities.
Reasonableness: A discontinuity is reasonable if the limit exists from both sides and matches the function value, otherwise it is not.
Analysis: Use limits to determine the nature of discontinuity at endpoints or where the function definition changes.
Example: At and for the above piecewise function, check left and right limits and function values.
The Tangent Line and Derivative
Secant and Tangent Lines
The tangent line to a curve at a point represents the instantaneous rate of change (the derivative) at that point. The secant line connects two points on the curve and its slope approximates the derivative as the points get closer.
Secant Slope:
Tangent Slope:
Tangent Line Equation:
Example: Find the slope of the tangent to at using the limiting value of the secant slope.
Summary Table: Types of Discontinuity
Type | Description | Example |
|---|---|---|
Removable | Limit exists, but function value is different or undefined | at |
Jump | Left and right limits exist but are not equal | Piecewise function with different values at a point |
Infinite | Function approaches infinity at a point | at |
Key Formulas
Limit Definition of Derivative:
Squeeze Theorem:
If for all near , and , then .
Additional info:
Some questions require interval notation and analysis of piecewise functions, which are common in introductory calculus.
Partial credit is awarded for showing work and justification, emphasizing the importance of process in calculus problem-solving.
