Question

The line \(y = L\) is a horizontal asymptote of the graph of the function \(f\) if

• A. \(\lim_{x \to +\infty} f(x) = L\) or \(\lim_{x \to -\infty} f(x) = L\)
• B. \(\lim_{x \to -\infty} f(x)\) do not exist
• C. \(\lim_{x \to +\infty} f(x)\) do not exist

Hint:

Remember: A horizontal asymptote is determined by the behavior of the function as \(x\) goes to \(\pm\infty\).
If the limit is a finite number \(L\), then \(y = L\) is a horizontal asymptote.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A horizontal asymptote is a horizontal line that the graph approaches as \(x\) tends to \(\pm\infty\).

Step 2: Key Approach:
The definition of a horizontal asymptote requires the limit of the function to be a finite number as \(x\) approaches infinity or negative infinity.

Step 3: Detailed Explanation:
Option (A) states: \(\lim_{x \to +\infty} f(x) = L\) or \(\lim_{x \to -\infty} f(x) = L\).
This is the correct definition.
Option (B) says the limit does not exist, which is not an asymptote.
Option (C) also says the limit does not exist.
Thus, the correct answer is (A).
A horizontal asymptote can be approached from the right or left.
If both limits exist and are equal, it's a single horizontal asymptote.
If they are different, there are two horizontal asymptotes.