Question

A line $y = L$ is called a horizontal asymptote of the graph of a function $f(x)$ if

• A. $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$
• B. $\lim_{x \to 0} f(x) = L$
• C. $\lim_{x \to 0} f(x) = -L$
• D. $\lim_{x \to 0^+} f(x) = \frac{1}{L}$

Hint:

To find horizontal asymptotes of rational functions, simply calculate the limit of the function as $x \to \infty$ by evaluating the ratio of the highest-degree terms.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the coordinates tend to infinity.

Step 2: Detailed Explanation:
A horizontal asymptote represents the limiting behavior of a function as the independent variable $x$ grows arbitrarily large in the positive or negative direction.
Thus, the horizontal line $y = L$ is a horizontal asymptote of the function $f(x)$ if the function approaches $L$ as $x$ tends to positive or negative infinity:
\[ \lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L \]
Other choices involving limits at $x \to 0$ describe the behavior of the function near the origin, which is unrelated to horizontal asymptotes.

Step 3: Final Answer
The correct option is (A).