Question

Find the vertical asymptotes of \[ f(x)=\frac{-8}{x^2-4}. \]

• A. \(x = 0\)
• B. \(x = 4\)
• C. \(x = \pm 2\)
• D. \(x = \pm 8\)

Hint:

Vertical asymptotes are found by setting the denominator to zero.
Check that the numerator is not also zero at those points (which would indicate a hole).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Vertical asymptotes occur where the denominator of a rational function is zero
and the numerator is not zero at those points.

Step 2: Key Formula or Approach:
Set the denominator equal to zero and solve for \(x\).

Step 3: Detailed Explanation:
Given \(f(x) = \frac{-8}{x^2 - 4}\).
The denominator is \(x^2 - 4\).
Set \(x^2 - 4 = 0\): \[ x^2 = 4 \implies x = \pm 2. \]
At \(x = 2\) and \(x = -2\), the denominator is zero and the numerator is non-zero.
Thus, the vertical asymptotes are \(x = 2\) and \(x = -2\).
This is option (C).
Vertical asymptotes are vertical lines where the function tends to infinity.
The function is undefined at these points.
The graph approaches these lines but never touches them.