Lecture Notes: Infinite Limits and Vertical Asymptotes
Definition of Infinite Limits
Infinite Limit: Occurs when the dependent variable (y-value) becomes boundless (positive or negative infinity) as the independent variable (x-value) approaches a finite value.
Notation:
If as (x \to c), (f(x) \to \infty), write: (\lim_{x \to c} f(x) = \infty).
Limits of this form technically do not exist but are studied for insights into function behavior.
Conditions for Infinite Limits
Occur when direct substitution results in a non-zero numerator and a zero denominator.
Examples
Function(f(x) = \frac{1}{x}):
Approaching 0 from the right (e.g., (0.1, 0.01, 0.001)), the outputs grow large, tending towards infinity.
(\lim_{x \to 0^+} \frac{1}{x} = \infty)
(\lim_{x \to 0^-} \frac{1}{x} = -\infty)
Graphically, as x approaches 0, the curve approaches infinity on the right and negative infinity on the left.
Notational Conventions for Infinite Limits
(c/\text{small positive} = \infty)
(c/\text{small negative} = -\infty)
Normal sign rules apply for division.
Limit Strategies (up to Section 2.4)
Direct Substitution:
If it gives a real number, the limit exists.
If it results in (0/0) indeterminate form, other techniques are used.
If it results in (x/0) (where x is non-zero), it leads to infinite limits.
Vertical Asymptotes
Definition: A line (x = c) is a vertical asymptote if at least one limit as (x) approaches (c) is infinite.
Identification:
Occurs when the denominator of a function equals zero.
Verify by showing one of the limits as (\pm \infty).
Examples of Vertical Asymptotes
Function (f(x) = \frac{x+3}{x-2}):
(\lim_{x \to 2^-} = -\infty)
(\lim_{x \to 2^+} = \infty)
Function (f(x) = \frac{x^2 - 3x + 2}{x^3 - 2x^2}):
(\lim_{x \to 0^+} = -\infty)
Function (f(x) = \frac{1+2x}{x^2}):
(\lim_{x \to 0^+} = \infty)
Tangential Function (\tan(x)) at (x = \frac{\pi}{2}):
(\lim_{x \to \frac{\pi}{2}^+} \tan(x) = -\infty)
Confirms (x = \frac{\pi}{2}) as a vertical asymptote.
Summary
Infinite limits provide insights into how functions behave near points where they become unbounded, crucial for understanding function behavior and locating vertical asymptotes.
While technically non-existent, conventions are applied to study these limits.