IndietroStudy Notes: Limits, Squeeze Theorem, and Infinite Limits
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Limits
Computing Limits
The concept of a limit is fundamental in calculus and describes the behavior of a function as its input approaches a particular value. Limits are used to define derivatives and integrals, and are essential for understanding continuity and the behavior of functions near points of interest.
Definition: The limit of a function f(x) as x approaches a value a is written as .
Key Properties:
If f(x) approaches a single value L as x approaches a, then .
Limits can often be computed by direct substitution, factoring, rationalizing, or using special limit laws.
Example:
Squeeze Theorem
The Squeeze Theorem is a useful tool for finding limits of functions that are difficult to evaluate directly. It states that if a function is "squeezed" between two other functions that have the same limit at a point, then the original function must also have that limit.
Theorem Statement: If for all x near a (except possibly at a), and , then .
Example: To find , note that . Since , by the Squeeze Theorem, .
Infinite Limits
Limits Approaching Infinity
An infinite limit occurs when the values of a function increase or decrease without bound as the input approaches a certain value. This is often associated with vertical asymptotes in the graph of a function.
Definition: means that f(x) grows arbitrarily large as x approaches a.
Vertical Asymptotes: If or , then x = a is a vertical asymptote of f(x).
Example: and
Summary Table: Types of Limits
The following table summarizes the main types of limits discussed:
Type of Limit | Definition | Example |
|---|---|---|
Finite Limit | Function approaches a finite value L as x approaches a | |
Infinite Limit | Function increases/decreases without bound as x approaches a | |
Squeeze Theorem | Function is bounded between two functions with the same limit |
