๐ Topic Summary
Infinite limits describe the behavior of a function as the input ($x$) approaches a specific value or infinity, resulting in the function's value growing without bound (approaching positive or negative infinity). Vertical asymptotes occur at $x$-values where the function approaches infinity or negative infinity. Identifying these asymptotes helps understand where a function is undefined and how it behaves near those points. In simpler terms, a vertical asymptote is like an invisible line the function gets closer and closer to, but never quite touches.
๐ Part A: Vocabulary
Match each term with its correct definition:
| Term |
Definition |
| 1. Infinite Limit |
A. A line $x=a$ where the function approaches $\pm \infty$ as $x$ approaches $a$. |
| 2. Vertical Asymptote |
B. The behavior of a function as $x$ approaches a value where the function's value increases or decreases without bound. |
| 3. Limit Notation |
C. $\lim_{x \to a} f(x) = \infty$ or $\lim_{x \to a} f(x) = -\infty$ |
| 4. Undefined |
D. A value in the domain of a function for which the function is not defined (e.g., division by zero). |
| 5. Rational Function |
E. A function that can be written as the ratio of two polynomials. |
โ๏ธ Part B: Fill in the Blanks
A ________ ________ occurs when the limit of a function as $x$ approaches a certain value is either positive or negative ________. This often happens when the denominator of a ________ function approaches ________, causing the function to become ________.
๐ค Part C: Critical Thinking
Explain, in your own words, how finding vertical asymptotes can help you graph a rational function. Provide an example to illustrate your explanation.