📚 Topic Summary
Limits at infinity describe the behavior of a function as the input (x) grows without bound (approaches positive or negative infinity). We're interested in what value, if any, the function approaches. A horizontal asymptote is a horizontal line that the graph of the function approaches as $x$ approaches positive or negative infinity. It visually represents the limit at infinity. Basically, we are looking to see what happens to the y-value of our function as x gets super big (positive or negative). It often involves comparing the degrees of the polynomials in the numerator and denominator of a rational function. Finding these limits and asymptotes are crucial in understanding the end behavior of functions.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Limit at Infinity |
A. A line that the function approaches as x approaches infinity. |
| 2. Horizontal Asymptote |
B. The behavior of a function as x approaches infinity. |
| 3. Degree of a Polynomial |
C. The highest power of the variable in the polynomial. |
| 4. Rational Function |
D. A function that can be written as a ratio of two polynomials. |
| 5. End Behavior |
E. The trend of a function's values as the input approaches positive and negative infinity. |
Answers: 1-B, 2-A, 3-C, 4-D, 5-E
✍️ Part B: Fill in the Blanks
As $x$ approaches infinity, the function $f(x) = \frac{1}{x}$ approaches _____. Therefore, the horizontal asymptote of $f(x)$ is $y = $ _____. If the degree of the numerator is _____ than the degree of the denominator in a rational function, the horizontal asymptote is y = 0. If the degrees are _____, the horizontal asymptote is the ratio of the leading _____. When the degree of the numerator is larger than the denominator, then there is no ____ asymptote.
Answers: 0, 0, less, equal, coefficients, horizontal
🤔 Part C: Critical Thinking
Explain in your own words how to determine the horizontal asymptote of a rational function. Give a specific example.
Answer: To find the horizontal asymptote, you need to consider the degrees of the numerator and denominator. If the degree of the denominator is greater than the numerator, the horizontal asymptote is y=0. If the degrees are equal, it's y = (leading coefficient of numerator) / (leading coefficient of denominator). If the degree of the numerator is greater, there's no horizontal asymptote. Example: For $f(x) = \frac{2x^2 + 1}{x^2 + 3}$, the degrees are equal (both 2), so the horizontal asymptote is y = 2/1 = 2.