Question

Vertical and horizontal asymptotes of the curve \(y = \frac{3x + 5}{7 - x}\) are respectively:

• A. \(x = 5/3, y = 7\)
• B. \(x = 7, y = 3\)
• C. \(x = 7, y = -3\)
• D. \(x = 7, y = -5/3\)

Hint:

Exam Tip:
For rational functions:

• Vertical asymptotes occur where denominator is zero (and numerator is not zero at that point).
• Horizontal asymptotes depend on the degrees of numerator and denominator.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the vertical and horizontal asymptotes of a rational function.

Step 2: Key Formula or Approach:

• Vertical asymptote: Set the denominator to zero and solve for \(x\).
• Horizontal asymptote: Look at the behavior as \(x \to \pm \infty\). If the degrees of numerator and denominator are equal, the horizontal asymptote is \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\).

Step 3: Detailed Explanation:
Given \(y = \frac{3x + 5}{7 - x}\).
• Vertical asymptote:
Set denominator to zero: \(7 - x = 0 \Rightarrow x = 7\).
• Horizontal asymptote:
As \(x \to \infty\), the function behaves like \(\frac{3x}{-x} = -3\).
So, \(y = -3\) is the horizontal asymptote. Thus, vertical asymptote is \(x = 7\) and horizontal asymptote is \(y = -3\).

Step 4: Final Answer:
This matches option (C). Therefore, option (C) is correct.