IndietroCalculus I: Infinite Limits and Vertical Asymptotes
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Chapter 2: Limits
Infinite Limits
Infinite limits describe the behavior of a function as its values increase or decrease without bound near a particular point. This concept is essential for understanding discontinuities and asymptotic behavior in calculus.
Definition: If f(x) grows arbitrarily large for all x sufficiently close (but not equal) to a, we write: and say the limit of f(x) as x approaches a is infinity.
Negative Infinite Limit: If f(x) is negative and grows arbitrarily large in magnitude for all x sufficiently close (but not equal) to a, we write: and say the limit of f(x) as x approaches a is negative infinity.
Non-existence: In both cases, the limit does not exist in the traditional sense, but describes unbounded behavior.
One-Sided Infinite Limits
One-sided infinite limits consider the behavior of a function as x approaches a value from only one side (left or right).
Right-Sided Infinite Limit: If f(x) becomes arbitrarily large for all x sufficiently close to a with x > a, we write:
Left-Sided Infinite Limit: Analogously, for x < a:
Negative infinite limits are written similarly, replacing with .
Vertical Asymptotes
A vertical asymptote occurs at x = a if the function increases or decreases without bound as x approaches a from either side.
Definition: If , , or , then the line x = a is a vertical asymptote of f.
Examples and Applications
Example: Evaluating Infinite Limits
Example 1:
Example 2:
Example: One-Sided Infinite Limits
is undefined (since the left and right limits are not equal).
Example: Determining Limits Analytically
Example 3:
Example 4:
Example 5:
Example 6:
Example 7:
Example 8:
Example: Location of Vertical Asymptotes
Let . Find the vertical asymptotes and evaluate the following limits:
Vertical asymptotes occur where the denominator is zero and the numerator is not zero, i.e., at and .
Summary Table: Infinite Limits and Vertical Asymptotes
Type of Limit | Symbolic Form | Behavior |
|---|---|---|
Infinite Limit | f(x) increases without bound as x approaches a | |
Negative Infinite Limit | f(x) decreases without bound as x approaches a | |
Right-Sided Infinite Limit | f(x) increases without bound as x approaches a from the right | |
Left-Sided Infinite Limit | f(x) increases without bound as x approaches a from the left | |
Vertical Asymptote | x = a | Occurs if any one-sided or two-sided limit is infinite at x = a |
Additional info: The above notes expand on the definitions and examples provided in the slides, adding context and a summary table for clarity. The images included directly illustrate infinite limits and vertical asymptotes as discussed in the text.
