BackInfinite Limits and Their Properties
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Limits & Continuity
Infinite Limits
In calculus and precalculus, infinite limits describe the behavior of a function as the input approaches a certain value and the output increases or decreases without bound. This concept is essential for understanding vertical asymptotes and the unbounded growth of functions near certain points.
Definition: Let f be a function defined on both sides of a, except possibly at a itself. We write: if the values of f(x) can be made arbitrarily large (positive) by taking x sufficiently close to a (but not equal to a).
Left and Right Infinite Limits:
These indicate the function grows without bound in the negative or positive direction as x approaches 0 from the left or right, respectively.
Examples of Infinite Limits
Example 1: As x approaches 2, the denominator approaches zero, causing the function to increase without bound.
Example 2: As x approaches 0, the denominator becomes very small, and the function grows very large in the positive direction.
General Property of Infinite Limits
For any real number n (where n is even): This holds because raising a small number to an even power keeps it positive, and the reciprocal grows without bound as the base approaches zero.
Special Cases and Observations
Observation: (This is a classic limit, not infinite, but often discussed in the context of limits approaching zero.)
Additional info: Infinite limits are closely related to the concept of vertical asymptotes in graphing rational functions. When a function approaches infinity as x approaches a certain value, the graph of the function will have a vertical asymptote at that value.
